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Cell_Biology_Alberts. Figure 8–61 GFPs that fluoresce at different wavelengths help reveal the connections that individual neurons make within the brain. This image shows differently colored neurons in one region of a mouse brain. The neurons randomly express different combinations of differently colored GFPs (see Figure 9–13), making it possible to distinguish and trace many individual neurons within a population. These images were obtained by genetically engineering the genes for four different fluorescent proteins, each flanked by loxP sites of recombination (see Figure 5–66), and integrating them into the mouse germ line. When crossed to a mouse that produced the Cre recombinase in neuronal cells, the fluorescent protein genes were randomly excised, producing neurons that express many different combinations of the four fluorescent proteins. Over 100 combinations of fluorescent protein can be produced, allowing scientists to distinguish one neuron from the next. The stunning appearance of these labeled


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Cell_Biology_Alberts. of the four fluorescent proteins. Over 100 combinations of fluorescent protein can be produced, allowing scientists to distinguish one neuron from the next. The stunning appearance of these labeled neurons has earned these animals the colorful nickname “brainbow mice.” (From J. Livet et al., Nature 450:56–62, 2007. With permission from Macmillan Publishers Ltd.)


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Cell_Biology_Alberts. Figure 8–62 In situ hybridization to mRNAs has been used to generate an atlas of gene expression in the mouse brain. This computer-generated image shows the expression of several different mRNAs specific to an area of the brain associated with learning and memory. Similar maps of expression patterns of all known genes in the mouse brain are compiled in the brain atlas project, which is available online. (From M. Hawrylycz et al., PLoS Comput. Biol. 7:e1001065, 2011.) Figure 8–63 RNA levels can be measured by quantitative RT-PCR. The fluorescence measured is generated by a dye that fluoresces only when bound to the double-stranded DNA products of the RT-PCR (see Figure 8–36). The red sample has a higher concentration of the mRNA being measured than does the blue sample, since it requires fewer PCR cycles to reach the same half-maximal concentration of double-stranded DNA. Based on this difference, the relative amounts of the mRNA in the two samples can be precisely determined.


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Cell_Biology_Alberts. that are expressed at reasonably high levels, and it is difficult to quantify (or even identify) rare RNAs. A more accurate method is based on the principles of PCR (Figure 8–63). Called quantitative RT-PCR (reverse transcription–polymerase chain reaction), this method begins with the total population of RNA molecules purified from a tissue or a cell culture. It is important that no DNA be present in the preparation; it must be purified away or enzymatically degraded. Two DNA primers that specifically match the mRNA of interest are added, along with reverse transcriptase, DNA polymerase, and the four deoxyribonucleoside triphosphates needed for DNA synthesis. The first round of synthesis is the reverse transcription of the RNA into DNA using one of the primers. Next, a series of heating and cooling cycles allows the amplification of that DNA strand by PCR (see Figure 8–36). The quantitative part of this method relies on a direct relationship between the rate at which the PCR product


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Cell_Biology_Alberts. and cooling cycles allows the amplification of that DNA strand by PCR (see Figure 8–36). The quantitative part of this method relies on a direct relationship between the rate at which the PCR product is generated and the original concentration of the mRNA species of interest. By adding chemical dyes to the PCR that fluoresce only when bound to double-stranded DNA, a simple fluorescence measurement can be used to track the progress of the reaction and thereby accurately deduce the starting concentration of the mRNA that is amplified. Although it seems complicated, this quantitative RT-PCR technique is relatively fast and simple to perform in the laboratory; it is currently the method of choice for accurately quantifying mRNA levels from any given gene.


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Cell_Biology_Alberts. Analysis of mRNAs by Microarray or RNA-seq Provides a Snapshot of Gene Expression As discussed in Chapter 7, a cell expresses only a subset of the many thousands of genes available in its genome; moreover, this subset differs from one cell type to another or, in the same cell, from one environment to the next. One way to determine which genes are being expressed by a population of cells or a tissue is to analyze which mRNAs are being produced.


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Cell_Biology_Alberts. The first tool that allowed investigators to analyze simultaneously the thousands of different RNAs produced by cells or tissues was the DNA microarray. Developed in the 1990s, DNA microarrays are glass microscope slides that contain hundreds of thousands of DNA fragments, each of which serves as a probe for the mRNA produced by a specific gene. Such microarrays allow investigators to monitor the expression of every gene in a genome in a single experiment. To do the analysis, mRNAs are extracted from cells or tissues and converted to cDNAs (see Figure 8–31). The cDNAs are fluorescently labeled and allowed to hybridize to the fragments bound to the microarray. An automated fluorescence microscope then determines which mRNAs were present in the original sample based on the array positions to which the cDNAs are bound (Figure 8–64).


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Cell_Biology_Alberts. Although microarrays are relatively inexpensive and easy to use, they suffer from one obvious drawback: the sequences of the mRNA samples to be analyzed must be known in advance and represented by a corresponding probe on the array. With the development of improved sequencing technologies, investigators increasingly use RNA-seq, discussed earlier, as a more direct approach for cataloging the RNAs produced by a cell. For example, this approach can readily detect alternative RNA splicing, RNA editing, and the many noncoding RNAs produced from a complex genome. DNA microarrays and RNA-seq analysis have been used to examine everything from the changes in gene expression that make strawberries ripen to the gene expression “signatures” of different types of human cancer cells; or from changes time (number of PCR cycles)


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Cell_Biology_Alberts. Figure 8–64 DNA microarrays are used to analyze the production of mRNA from mRNA from thousands of different mRNAs in a single experiment. In this example, sample 1 sample 2 mRNA is collected from two different cell samples—for example, cells treated with a hormone and untreated cells of the same type—to allow for a direct comparison of the specific genes expressed under both conditions. convert to cDNA, convert to cDNA,


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Cell_Biology_Alberts. The mRNAs are converted to cDNAs that are labeled with a red fluorescent labeled with red labeled with green dye for one sample and a green fluorescent dye for the other. The labeled samples are mixed and then allowed to hybridize to the microarray. Each microscopic spot on the microarray is a 50-nucleotide DNA molecule of defined sequence made by chemical synthesis and spotted on the array. The DNA sequence represented by each spot is different, and the hundreds of thousands of such spots are designed to span the sequence of the genome. The DNA sequence of each spot is kept track of by computer. After incubation, the array is washed and the fluorescence scanned. Only a small proportion of the microarray, representing 676 genes, is shown. Red spots indicate that the gene in sample 1 is expressed at a higher level than the corresponding gene in sample 2, and the green spots indicate the opposite. Yellow spots reveal genes that are expressed at about equal levels in both cell samples.


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Cell_Biology_Alberts. at a higher level than the corresponding gene in sample 2, and the green spots indicate the opposite. Yellow spots reveal genes that are expressed at about equal levels in both cell samples. The intensity of the fluorescence provides an estimate of how much RNA is present from a gene. Dark spots indicate little or no expression of the gene whose probe is located at that position in the array.


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Cell_Biology_Alberts. that occur as cells progress through the cell cycle to those made in response to sudden shifts in temperature. Indeed, because these approaches allow the simultaneous monitoring of large numbers of RNAs, they can detect subtle changes in a cell, changes that might not be manifested in its outward appearance or behavior.


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Cell_Biology_Alberts. Comprehensive studies of gene expression also provide information that is useful for predicting gene function. Earlier in this chapter, we discussed how identifying a protein’s interaction partners can yield clues about that protein’s function. A similar principle holds true for genes: information about a gene’s function can be deduced by identifying genes that share its expression pattern. Using an approach called cluster analysis, one can identify sets of genes that are coordinately regulated. Genes that are turned on or turned off together under different circumstances are likely to work in concert in the cell: they may encode proteins that are part of the same multiprotein machine, or proteins that are involved in a complex coordinated activity, such as DNA replication or RNA splicing. Characterizing a gene whose function is unknown by grouping it with known genes that share its transcriptional behavior is sometimes called “guilt by association.” Cluster analyses have been used to


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Cell_Biology_Alberts. Characterizing a gene whose function is unknown by grouping it with known genes that share its transcriptional behavior is sometimes called “guilt by association.” Cluster analyses have been used to analyze the gene expression profiles that underlie many interesting biological processes, including wound healing in humans (Figure 8–65).


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Cell_Biology_Alberts. Figure 8–65 Using cluster analysis to identify sets of genes that are coordinately regulated. Genes that have the same expression pattern are likely to be involved in common pathways or processes. To perform a cluster analysis, RNA-seq or microarray data are obtained from cell samples exposed to a variety of different conditions, and genes that show coordinate changes in their expression pattern are grouped together. In this experiment, human fibroblasts were deprived of serum for 48 hours; serum was then added back to the cultures at time 0 and the cells were harvested for microarray analysis at different time points. Of the 8600 genes depicted here (each represented by a thin, vertical line), just over 300 showed threefold or greater variation in their expression patterns in response to serum reintroduction. Here, red indicates an increase in expression; green is a decrease in expression. On the basis of the results of many other experiments, the 8600 genes have been grouped in


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Cell_Biology_Alberts. to serum reintroduction. Here, red indicates an increase in expression; green is a decrease in expression. On the basis of the results of many other experiments, the 8600 genes have been grouped in clusters based on similar patterns of expression. The results of this analysis show that genes involved in wound healing are turned on in response to serum, while genes involved in regulating cell-cycle progression and cholesterol biosynthesis are shut down. (From M.B. Eisen et al., Proc. Natl Acad. Sci. USA 94:14863–14868, 1998. With permission from National Academy of Sciences.)


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Cell_Biology_Alberts. Figure 8–66 Chromatin immunoprecipitation. This method allows the identification of all the sites in a genome that a transcription regulator occupies in vivo. The identities of the precipitated, amplified DNA fragments are determined by DNA sequencing. Genome-wide Chromatin Immunoprecipitation Identifies Sites on the Genome Occupied by Transcription Regulators


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Cell_Biology_Alberts. We have discussed several strategies to measure the levels of individual RNAs in a cell and to monitor changes in their levels in response to external signals. But this information does not tell us how such changes are brought about. We saw in Chapter 7 that transcription regulators, by binding to cis-regulatory sequences in DNA, are responsible for establishing and changing patterns of transcription. Typically, these proteins do not occupy all of their potential cis-regulatory sequences in the genome under all conditions. For example, in some cell types, the regulatory protein may not be expressed, or it may be present but lack an obligatory partner protein, or it may be excluded from the nucleus until an appropriate signal is received from the cell’s environment. Even if the protein is present in the nucleus and is competent to bind DNA, other transcription regulators or components of chromatin can occupy overlapping DNA sequences and thereby occlude some of its cis-regulatory


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Cell_Biology_Alberts. is present in the nucleus and is competent to bind DNA, other transcription regulators or components of chromatin can occupy overlapping DNA sequences and thereby occlude some of its cis-regulatory sequences in the genome.


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Cell_Biology_Alberts. Chromatin immunoprecipitation provides a way to experimentally determine all the cis-regulatory sequences in a genome that are occupied by a given transcription regulator under a particular set of conditions (Figure 8–66). In this approach, proteins are covalently cross-linked to DNA in living cells, the cells are broken open, and the DNA is mechanically sheared into small fragments. Antibodies directed against a given transcription regulator are then used to purify the DNA that became covalently cross-linked to that protein in the cell. This DNA is then sequenced using the rapid methods discussed earlier; the precise location of each precipitated DNA fragment along the genome is determined by comparing its DNA sequence to that of the whole genome sequence (Figure 8–67). In this way, all of the sites occupied by the transcription regulator in the cell sample can be mapped across the cell’s genome (see Figure 7–37). In combination with microarray or RNA-seq information, chromatin


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Cell_Biology_Alberts. all of the sites occupied by the transcription regulator in the cell sample can be mapped across the cell’s genome (see Figure 7–37). In combination with microarray or RNA-seq information, chromatin immunoprecipitation can identify the key transcriptional regulator responsible for specifying a particular pattern of gene expression.


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Cell_Biology_Alberts. Chromatin immunoprecipitation can also be used to deduce the cis-regulatory sequences recognized by a given transcription regulator. Here, all the DNA sequences precipitated by the regulator are lined up (by computer) and features in common are tabulated to produce the spectrum of cis-regulatory sequences recognized by the protein (see Figure 7–9A). Chromatin immunoprecipitation is also used routinely to identify the positions along a genome that are bound by the various types of modified histones discussed in Chapter 4. In this case, antibodies specific to the particular histone modification are employed (see Figure 8–67). A variation of the technique can also be used to map positions of chromosomes that are in physical proximity (see Figure 4–48). Ribosome Profiling Reveals Which mRNAs Are Being Translated in the Cell


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Cell_Biology_Alberts. Ribosome Profiling Reveals Which mRNAs Are Being Translated in the Cell In preceding sections, we discussed several ways that RNA levels in the cell can be monitored. But for mRNAs, this represents only one step in gene expression, and we are often more interested in the final level of the protein produced by the gene. As described in the first part of this chapter, mass-spectroscopy methods can be used to monitor the levels of all proteins in the cell, including modified forms of the proteins. However, if we want to understand how synthesis of proteins is controlled by the cell, we need to consider the translation step of gene expression.


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Cell_Biology_Alberts. An approach called ribosome profiling provides an instantaneous map of the position of ribosomes on each mRNA in the cell and thereby identifies those many other DNA fragments that comprise the rest of the genome mRNAs that are being actively translated. To accomplish this, total RNA from a cell line or tissue is exposed to RNAses under conditions where only those RNA sequences covered by ribosomes are spared. The protected RNAs are released from ribosomes, converted to DNA, and the nucleotide sequence of each is determined (Figure 8–68). When these sequences are mapped on the genome, the position of ribosomes across each mRNA species can be ascertained.


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Cell_Biology_Alberts. Ribosome profiling has revealed many cases where mRNAs are abundant but are not translated until the cell receives an external signal. It has also shown that many open reading frames (ORFs) that were too short to be annotated as genes are actively translated and probably encode functional, albeit very small, proteins (Figure 8–69). Finally, ribosome profiling has revealed the ways that cells rapidly and globally change their translation patterns in response to sudden changes in temperature, nutrient availability, or chemical stress.


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Cell_Biology_Alberts. We have seen that nucleic acid methodologies developed in the past 40 years have completely changed the way that cell and molecular biology is studied. But they have also had a profound effect on our day-to-day lives. Many human pharmaceuticals in routine use (insulin, human growth hormone, blood-clotting factors, and interferon, for example) are based on cloning human genes and expressing the encoded proteins in large amounts. As DNA sequencing continues to drop in cost, more and more individuals will elect to have their genome sequenced; this information can be used to predict susceptibility to diseases (often with the option of minimizing this possibility by appropriate behavior) or to predict the way an individual will respond to a given drug. The genomes of tumor cells from an individual can be sequenced to determine the best type of anticancer treatment. And mutations that cause or greatly increase the risk of disease continue to be identified at an unprecedented pace. Using the


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Cell_Biology_Alberts. can be sequenced to determine the best type of anticancer treatment. And mutations that cause or greatly increase the risk of disease continue to be identified at an unprecedented pace. Using the recombinant DNA technologies discussed in this chapter, these mutations can then be introduced into animals, such as mice, that can be studied in the laboratory. The resulting transgenic animals, which often mimic some of the phenotypic abnormalities associated with the condition in patients, can be used to explore the cellular and molecular basis of the disease and to screen for drugs that could potentially be used therapeutically in humans.


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number of sequence reads

Cell_Biology_Alberts. number of sequence reads


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Cell_Biology_Alberts. Figure 8–67 Results of several chromatin immunoprecipitations showing proteins bound to the control region that control expression of the Oct4 gene. In this series of chromatin immunoprecipitation experiments, antibodies directed against a transcription regulator (first three panels) or a particular histone modification (fourth panel) were used to precipitate bound, cross-linked DNA. Precipitated DNA was sequenced, and the positions across the genome were mapped. (Only the small part of the mouse genome containing the Oct4 gene is shown.) The results show that, in the embryonic stem cells analyzed in these experiments, Oct4 binds upstream of its own gene and that Sox2 and Nanog are bound in close proximity. Oct4, Sox2, and Nanog are key regulators in embryonic stem cells (discussed in Chapter 22) and this experiment reveals the position on the genome through which they exert their effects on Oct4 expression. In the fourth panel, the positions of a histone modification associated with


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Cell_Biology_Alberts. 22) and this experiment reveals the position on the genome through which they exert their effects on Oct4 expression. In the fourth panel, the positions of a histone modification associated with actively transcribed genes is shown (see Figure 4–39). Finally, the bottom panel shows the RNA produced from the Oct4 gene under the same conditions used for the chromatin immunoprecipitations. Note that the introns and exons are relatively easy to identify from these RNA-seq data.


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Cell_Biology_Alberts. Figure 8–68 Ribosome profiling. RNA AUG UGA AAAAAAAAAAAA is purified from cells and digested with an RNAse to leave only those portions AUG of the mRNAs that are protected by a bound ribosome. These short pieces remove ribosomes convert RNA to DNA and sequence Although we tend to think of recombinant DNA research in terms of animal biology, these techniques have also had a profound impact on the study of plants. In fact, certain features of plants make them especially amenable to recombinant DNA methods.


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Cell_Biology_Alberts. When a piece of plant tissue is cultured in a sterile medium containing nutrients and appropriate growth regulators, some of the cells are stimulated to proliferate indefinitely in a disorganized manner, producing a mass of relatively undifferentiated cells called a callus. If the nutrients and growth regulators are carefully manipulated, one can induce the formation of a shoot within the callus, and in many species a whole new plant can be regenerated from such shoots. In a number of plants—including tobacco, petunia, carrot, potato, and Arabidopsis—a single cell from such a callus (known as a totipotent cell) can be grown into a small clump of cells from which a whole plant can be regenerated (see Figure 7–2B). Just as mutant mice can be derived by the genetic manipulation of embryonic stem 200 ORF discovered through ribosome profling nucleotide pairs codes for a protein of 20 amino acids of protected RNA (approximately 20 nucleotides in length) are converted to DNA and sequenced.


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Cell_Biology_Alberts. stem 200 ORF discovered through ribosome profling nucleotide pairs codes for a protein of 20 amino acids of protected RNA (approximately 20 nucleotides in length) are converted to DNA and sequenced. The resulting information is displayed as the number of sequence reads along each position of the genome. In the diagram here, the data for only one gene, whose mRNA is being efficiently translated, are shown. Ribosome profiling provides this type of information for every mRNA produced by the cell.


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Cell_Biology_Alberts. Figure 8–69 Ribosome profiling can identify new genes. This experiment shows the discovery of a previously unrecognized gene—one that encodes a protein of only 20 amino acids. At the top is shown a portion of a viral genome with two previously annotated genes. Below are the results of a ribosome profiling experiment, displayed across the same section of the genome, after the virus was infected into human cells. The results show that the left-hand gene is not expressed under these conditions, the right-hand gene is expressed at low levels, and a previously unrecognized gene that lies between them position along genome is expressed at high levels. 508 Chapter 8: Analyzing Cells, Molecules, and Systems leaf discs incubated with genetically engineereddiscs removed from cells in culture, so transgenic plants can be created from plant cells transfected with DNA in culture (Figure 8–70).


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Cell_Biology_Alberts. The ability to produce transgenic plants has greatly accelerated progress in many areas of plant cell biology. It has played an important part, for example, in isolating receptors for growth regulators and in analyzing the mechanisms of morphogenesis and of gene expression in plants. These techniques have also opened up many new possibilities in agriculture that could benefit both the farmer and the consumer. They have made it possible, for example, to modify the ratio of lipid, starch, and protein in seeds, to impart pest and virus resistance to plants, and to create modified plants that tolerate extreme habitats such as salt marshes or water-stressed soil. One variety of rice has been genetically engineered to produce β-carotene, the precursor of vitamin A. Were it to replace conventional rice, this “golden rice”—so-called because of its faint yellow color—could help to alleviate severe vitamin A deficiency, which causes blindness in hundreds of thousands of children in the


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Cell_Biology_Alberts. rice, this “golden rice”—so-called because of its faint yellow color—could help to alleviate severe vitamin A deficiency, which causes blindness in hundreds of thousands of children in the developing world each year.


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Cell_Biology_Alberts. Genetics and genetic engineering provide powerful tools for understanding the function of individual genes in cells and organisms. In the classical genetic approach, random mutagenesis is coupled with screening to identify mutants that are deficient in a particular biological process. These mutants are then used to locate and study the genes responsible for that process.


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Cell_Biology_Alberts. Gene function can also be ascertained by reverse genetic techniques. DNA engineering methods can be used to alter genes and to re-insert them into a cell’s chromosomes so that they become a permanent part of the genome. If the cell used for this gene transfer is a fertilized egg (for an animal) or a totipotent plant cell in culture, transgenic organisms can be produced that express the mutant gene and pass it on to their progeny. Especially important for cell and molecular biology is the ability to alter cells and organisms in highly specific ways—allowing one to discern the effect on the cell or the organism of a designed change in a single protein or RNA molecule. For example, genomes can be altered so that the expression of any gene can be switched on or off by the experimenter.


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Cell_Biology_Alberts. Figure 8–70 Transgenic plants can be made using recombinant DNA techniques optimized for plants. A disc is cut out of a leaf and incubated in a culture of Agrobacterium that carries a recombinant plasmid with both a selectable marker and a desired genetically engineered gene. The wounded plant cells at the edge of the disc release substances that attract the bacteria, which inject their DNA into the plant cells. Only those plant cells that take up the appropriate DNA and express the selectable marker gene survive and proliferate and form a callus. The manipulation of growth factors supplied to the callus induces it to form shoots, which subsequently root and grow into adult plants carrying the engineered gene.


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Cell_Biology_Alberts. Many of these methods are being expanded to investigate gene function on a genome-wide scale. The generation of mutant libraries in which every gene in an organism has been systematically deleted, disrupted, or made controllable by the experimenter provides invaluable tools for exploring the role of each gene in the elaborate molecular collaboration that gives rise to life. Technologies such as RNA-seq and DNA microarrays can monitor the expression of tens of thousands of genes simultaneously, providing detailed, comprehensive snapshots of the dynamic patterns of gene expression that underlie complex cell processes.


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Cell_Biology_Alberts. Quantitative experiments combined with mathematical theory mark the beginning of modern science. Galileo, Kepler, Newton, and their contemporaries did more than set out some rules of mechanics and offer an explanation of the movements of the planets around the Sun: they showed how a quantitative mathematical approach could provide a depth and precision of understanding, at least for physical systems, that had never before been dreamed to be possible. What is it that gives mathematics this almost magical power to explain the natural world, and why has mathematics played so much more important a part in physical sciences than in biology? What do biologists need to know about mathematics?


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Cell_Biology_Alberts. Mathematics can be viewed as a tool for deriving logical consequences from propositions. It differs from ordinary intuitive reasoning in its insistence on rigorous, accurate logic and the precise treatment of quantitative information. If the initial propositions are correct, then the deductions drawn from them by mathematics will be true. The surprising power of mathematics comes from the length of the chains of reasoning that rigorous logic and mathematical arguments make possible, and from the unexpectedness of the conclusions that can be reached, often revealing connections that one would not otherwise have guessed at. Reversing the argument, mathematics provides a way to test experimental hypotheses: if mathematical reasoning from a given hypothesis leads to a prediction that is not true, then the hypothesis is not true.


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Cell_Biology_Alberts. Clearly, mathematics is not much use unless we can frame our ideas—our initial hypotheses—about the given system in a precise, quantitative form. A mathematical edifice raised on a rickety or—even worse—a vague or overcomplicated set of propositions is likely to lead us astray. For mathematics to be useful, we must focus our analysis on simple subsystems in which we can pick out key quantitative parameters and frame well-defined hypotheses. This approach has been used with great success in physics for centuries, but it has been less common in biology. But times are changing, and more and more it is becoming possible for biologists to exploit the power of quantitative mathematical analysis.


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Cell_Biology_Alberts. In this final section of our methods chapter, we do not attempt to teach readers every way in which mathematics can be fruitfully applied to biological problems. Rather, we simply aim to give a sense of what mathematics and quantitative approaches can do for us in modern biology. We focus primarily on the important principles that mathematics teaches us about the dynamics of molecular interactions, and how mathematics can unveil surprising and useful features of complex systems containing feedback. We will illustrate these principles using the regulation of gene expression by transcription regulators like those discussed in Chapter 7. The same principles apply to the post-transcriptional regulatory systems that govern cell signaling (Chapter 15), cell-cycle control (Chapter 17), and essentially all cell processes.


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Cell_Biology_Alberts. Cell function and regulation depend on transient interactions among thousands of different macromolecules in the cell. We often summarize these interactions in this book with schematic cartoons. These diagrams are useful, but a complete picture requires a deeper, more quantitative level of understanding. To meaningfully assess the biological impact of any interaction in the cell, we need to know in precise terms how the molecules interact, how they catalyze reactions, and, most importantly, how the behaviors of the molecules change over time. If a cartoon shows that protein A activates protein B, for example, we cannot judge the importance of this relationship without quantitative details about the concentrations, affinities, and kinetic behaviors of proteins A and B.


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Cell_Biology_Alberts. Let us begin by defining two different types of regulatory interaction in our cartoons: one designating inhibition and the other designating activation. If the protein product of gene X is a transcription repressor that inhibits the expression of gene Z, we depict the relationship as a red bar-headed line ( ) drawn between genes X and Z (Figure 8–71). If the protein product of gene Y is a transcription activator that induces the expression of gene Z, then a green arrow ( ) is drawn between genes Y and Z.


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Cell_Biology_Alberts. The regulation of one gene’s expression by another is more complicated than a single arrow connecting them, and a complete understanding of this regulation requires that we tease apart the underlying biochemical processes. Figure 8–72A sketches some of the biochemical steps in the activation of gene expression by a transcription activator. A gene encoding the activator, designated as gene A, will produce its product, protein A, via an RNA intermediate. This protein A will then bind to pX, the regulatory promoter of gene X, to form the complex A:pX. Once the A:pX complex forms, it stimulates the production of an RNA transcript that is subsequently translated to produce protein X.


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Cell_Biology_Alberts. We will focus here on the binding interaction that lies at the heart of this regulatory system: the interaction between protein A and the promoter pX. Any molecule of protein A that is bound to pX can also dissociate from it. The steps represented by the green activation arrow in Figure 8–72A include both the binding of A to pX and the dissociation of the complex A:pX to re-form A and pX, as illustrated A:pX rate of complex formation = k [A][pX] rate of complex dissociation = koff [A:pX] at steady state: on [A][pX] = koff [A:pX] GENE X GENE Y Figure 8–71 Diagrams that summarize biochemical relationships. Here, a simple cartoon indicates that gene X represses gene Z (left) whereas gene Y activates gene Z (right).


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Cell_Biology_Alberts. Figure 8–72 A simple transcriptional interaction. (A) Genes A and X each produce a protein, with the product of gene A serving as a transcription activator to stimulate expression of gene X. As indicated by the green arrow, stimulation depends in part on the binding of protein A to the promoter region of gene X, designated as pX. (B) The binding of protein A to the gene promoter is determined by the concentrations of the two binding partners (denoted as [A] and [pX], in units of mol/liter, or M), the association rate constant kon (in units of sec–1 M–1), and the dissociation rate constant koff (in units of sec–1). (C) At steady state, the rates of association and dissociation are equal, and the concentration of the bound complex is determined by Equation 8–1, in which the two rate constants are combined in the equilibrium constant K. (D) Equation 8–2 can be derived to calculate the steady-state concentration of bound complex at a known total concentration of the promoter [pXT]. (E)


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Cell_Biology_Alberts. are combined in the equilibrium constant K. (D) Equation 8–2 can be derived to calculate the steady-state concentration of bound complex at a known total concentration of the promoter [pXT]. (E) Rearrangement of Equation 8–2 yields Equation 8–3, which allows calculation of the fraction of promoter pX that is occupied by protein A.


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Cell_Biology_Alberts. by the notation in Figure 8–72B. This reaction notation is more informative than the diagrams in our figures, but has its own limitations. Suppose that the concentration of A increases by a factor of ten as a response to an environmental input. If A increases, we intuitively know that A:pX should increase too, but we cannot determine the amount of the increase without additional information. We need to know the affinity of the binding interaction and the concentrations of the components. With this information in hand, we can rigorously derive the answer.


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Cell_Biology_Alberts. As discussed earlier and in Chapter 3 (see Figure 3–44), we know that the formation of a complex between two binding partners, such as A and pX, depends on a rate constant kon, which describes how many productive collisions occur per unit time per protein at a given concentration of pX. The rate of complex formation equals the product of this rate constant kon and the concentrations of A and pX (see Figure 8–72B). Complex dissociation occurs at a rate koff multiplied by the concentration of the complex. The rate constant koff can differ by orders of magnitude for different DNA sequences because it depends on the strength of the non-covalent bonds formed between A and pX.


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Cell_Biology_Alberts. We are primarily interested in understanding the amount of bound promoter complex at equilibrium or steady state, where the rate of complex formation equals the rate of complex dissociation. Under these conditions, the concentration of the promoter complex is specified by a simple equation that combines the two rate constants into a single equilibrium constant K = kon/koff (Equation 8–1; Figure 8–72C). K is sometimes called the association constant, Ka. The larger this constant K, the stronger the interaction between A and pX (see Figure 3–44). The reciprocal of K is the dissociation constant, Kd.


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Cell_Biology_Alberts. To calculate the steady-state concentration of promoter complex using Equation 8–1, we need to account for another complication: both A and pX exist in two forms—free in solution and bound to each other. In most cases, we know the total concentration of pX and not the free or bound concentrations, so we must find a way to use the total concentration in our calculations. To do this, we first specify that the total concentration of pX ([pXT ]) is the sum of the concentrations of free ([pX]) and bound ([A:pX]) forms (Figure 8–72D). This leads to a new equation that allows us to use [pXT ] to calculate the steady-state concentration of the promoter complex ([A:pX]) (Equation 8–2, Figure 8–72D).


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Cell_Biology_Alberts. Protein A also exists in two forms: free ([A]) and bound to pX ([A:pX]). In a cell, there are typically one or two copies of pX (assuming there is only one gene X per haploid genome) and multiple copies of A. As a result, we can safely assume that from the viewpoint of A, [A:pX] is negligible relative to the total [AT ]. This means that [A] ≈ [AT ], and we can just plug in the values of total [AT ] in Equation 8–2 without incurring appreciable error in the calculation of [A:pX]. Now, we are ready to determine the effects of increasing the concentration of


Cell_Biology_Alberts

Cell_Biology_Alberts. Now, we are ready to determine the effects of increasing the concentration of A. Suppose that K = 108 M–1, which is a typical value for many such interactions. The starting concentration of A is [AT ] = 10–9 M, and [pXT ] = 10–10 M (assuming there is one copy of gene X in a haploid yeast cell, for example, with a volume of around 2 × 10–14 L). Using Equation 8–2, we find that a tenfold increase in the concentration of A causes the amount of promoter complex [A:pX] to increase 5.5fold, from 0.09 × 10–10 M to 0.5 × 10–10 M at steady state. The effects of a tenfold increase in the concentration of A will vary dramatically depending on its starting concentration relative to the equilibrium constant. Only through this mathematical approach can we achieve a thorough understanding of what these effects will be and what impact they will have on the biological response.


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Cell_Biology_Alberts. To assess the biological impact of a change in transcription activator levels, it is also important in many cases to determine the fraction of the target gene promoter that is bound by the activator, since this number will be directly proportional to the activity of the gene’s promoter. In our case, we can calculate the fraction of the gene X promoter, pX, that has protein A bound to it by rearranging Equation 8–2 (Equation 8–3, Figure 8–72E). This fraction can be viewed as the probability that promoter pX is occupied, averaged over time. It is also equal to the average occupancy across a large population of cells at any instant in time. When there is no protein A present, pX is always free, the bound fraction is zero, and transcription is off. When [A] = 1/K, the promoter pX has a 50% chance of being occupied. When [A] greatly exceeds 1/K, the bound fraction is almost equal to one, meaning that pX is fully occupied and transcription is maximal.


Cell_Biology_Alberts

Cell_Biology_Alberts. The most important and basic insights for which we, as biologists, depend on mathematics concern the behavior of regulatory systems over time. This is the central theme of dynamics, and it was for the solution of problems in dynamics that the techniques of calculus were developed, by Newton and Leibniz, in the seventeenth century. Briefly, the general problem is this: if we are given the rates of change of a set of variables that characterize the system at any instant, how can we compute its future state? The problem becomes especially interesting, and the predictions often remarkable, when the rates of change themselves depend on the values of the state variables, as in systems with feedback.


Cell_Biology_Alberts

Cell_Biology_Alberts. Let us return to Equation 8–2 (Figure 8–72D), which tells us that when [A] changes, [A:pX] at steady state will also change to a new concentration that we can calculate with precision. However, [A:pX] does not change instantaneously to this value. If we hope to understand the behavior of this system in detail, we must also ask how long it takes [A:pX] to get to its new steady-state value inside the cell. Equation 8–2 cannot answer this question. We need calculus.


Cell_Biology_Alberts

Cell_Biology_Alberts. The most common strategy for solving this problem is to use ordinary differential equations. The equations that describe biochemical reactions have a simple premise: the rate of change in the concentration of any molecular species X (that is, d[X]/dt) is given by the balance of the rate of its appearance with that of its disappearance. For our example, the rate of change in the concentration of the bound promoter complex, [A:pX], is determined by the rates of complex assembly and disassembly. We can incorporate these rates into the differential equation shown in Figure 8–73A (Equation 8–4). When [A] changes, Equation 8–4 can be solved to generate the concentration of [A:pX] as a function of time. Notice that when kon [A][pX] = koff [A:pX], then d[A:pX]/dt = 0 and [A:pX] stops changing. At this point, the system has reached the steady state.


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Cell_Biology_Alberts. Calculation of all [A:pX] values as a function of time, using Equation 8–4, allows us to determine the rate at which [A:pX] reaches its steady-state value. Because this value is attained asymptotically, it is often most useful to compare the times needed to get to 50, 90, or 99 percent of this new steady state. The simplest way to determine these values is to solve Equation 8–4 with a method called numerical integration, which involves plugging in values for all of the parameters (kon, koff, etc.) and then using a computer to determine the values of [A:pX] over time, starting from given initial concentrations of [A] and [pX]. For kon = 0.5 × 107 sec–1 M–1, koff = 0.5 × 10–1 sec–1 (K = 108 M–1 as above), and [pXT ] = 10–10 M, it takes [A:pX] about 5, 20, and 40 seconds to reach 50, 90, and 99 percent of the new steady-state value following a sudden tenfold change in [A] (Figure 8–73B). Thus, a sudden jump in [A] does not have instantaneous effects, as we might have assumed from looking


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Cell_Biology_Alberts. 99 percent of the new steady-state value following a sudden tenfold change in [A] (Figure 8–73B). Thus, a sudden jump in [A] does not have instantaneous effects, as we might have assumed from looking at the cartoon in Figure 8–72A.


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Cell_Biology_Alberts. Differential equations therefore allow us to understand the transient dynamics of biochemical reactions. This tool is critical for achieving a deep understanding of cell behavior, in part because it allows us to determine the dependence of the dynamics inside cells on parameters that are specific to the particular molecules Figure 8–73 Using differential equations to study the dynamics and steady-state behavior of a biological system.


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Cell_Biology_Alberts. Figure 8–73 Using differential equations to study the dynamics and steady-state behavior of a biological system. (A) Equation 8–4 is an ordinary differential equation for calculating the rate of change in the formation of bound promoter complex in response to a change in other components. (B) Formation of [A:pX] after a tenfold increase in [A], as determined by solving Equation 8–4. In blue is the solution corresponding to kon = 0.5 × 107 sec–1 M–1 and koff = 0.5 × 10–1 sec–1. In this case, it takes [A:pX ] about 5, 20, and 40 seconds to reach 50, 90, and 99 percent of the new steady-state value. For the red curve, the kon and koff values are doubled, and the system reaches the same steady state more rapidly.


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Cell_Biology_Alberts. [A:pX] (multiples of initial value) involved. For example, if we double the values of both kon and koff, then Equa tion 8–1 (Figure 8–72C) indicates that the steady-state value of [A:pX] does not change. However, the time it takes to reach 50% of this steady state after a ten-fold 4.5 3.5 2.5 d[A:pX] = rate of complex formation – rate of complex dissociation 1.5 d[A:pX] 0.5 on [A][pX] – koff [A:pX] protein production rate = ˜.m change in [A] in our example changes from about 5 seconds to 2 seconds (see Figure 8–73B). These insights are not accessible from either cartoons or equilibrium equations. This is an unusually simple example; mathematical descriptions such as differential equations become more indispensible for understanding biological interactions as the number of interactions increases. Both Promoter Activity and Protein Degradation Affect the Rate of Change of Protein Concentration


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Cell_Biology_Alberts. Both Promoter Activity and Protein Degradation Affect the Rate of Change of Protein Concentration To understand our gene regulatory system further, we also need to describe the dynamics of protein X production in response to changes in the amount of transcription activator protein A. Here again, we use an ordinary differential equation for the rate of change of protein X concentration—determined by the balance of the rate of production of protein X through expression of gene X and the protein’s rate of degradation.


Cell_Biology_Alberts

Cell_Biology_Alberts. Let us begin with the rate of protein X production, which is determined primarily by the occupancy of the promoter of gene X by protein A. The binding and dissociation of a transcription regulator at a promoter generally occurs on a much faster time scale than transcription initiation, causing many binding and unbinding events to occur before transcription proceeds. As a result, we can assume that the binding reaction is at equilibrium on the time scale of transcription, and we can calculate promoter occupancy by protein A using the equilibrium equation discussed earlier (Equation 8–3, Figure 8–72E). To determine transcription rate, we simply multiply the occupied promoter fraction by a transcription rate constant, β, that represents the binding of RNA polymerase and the subsequent steps that lead to production of mRNA and protein (Figure 8–74A). If each mRNA molecule produces, on average, m molecules of protein product, then we can determine protein production rate by multiplying the


Cell_Biology_Alberts

Cell_Biology_Alberts. lead to production of mRNA and protein (Figure 8–74A). If each mRNA molecule produces, on average, m molecules of protein product, then we can determine protein production rate by multiplying the transcription rate by m (Figure 8–74A).


Cell_Biology_Alberts

Cell_Biology_Alberts. Now let us consider the factors that influence protein X degradation and its dilution due to cell growth. Degradation generally results in an exponential decline in protein levels, and the average time required for a specific protein to be 1.0 0.5 Figure 8–74 effect of protein lifetime on the timing of the response.


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Cell_Biology_Alberts. Figure 8–74 effect of protein lifetime on the timing of the response. (A) Equations for calculation of the rates of gene X transcription, protein X production, and protein X degradation, as explained in the text. (B) Equation 8–5 is an ordinary differential equation for calculating the rate of change in protein X in response to changes in other components. (C) When the rate of change in protein X is zero (steady state), its concentration can be calculated with Equation 8–6, revealing a direct relationship with protein lifetime (τ). (D) The solution of Equation 8–5 specifies the concentration of protein X over time as it approaches its steady-state concentration. (E) Response time depends on protein lifetime. As described in the text, the time that it takes a protein to reach a new steady state is greater when the protein is more stable. Here, the blue line corresponds to a protein with a lifetime that is 2.5-fold shorter than the lifetime of the protein in red.


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Cell_Biology_Alberts. degraded is defined as its mean lifetime, τ. In our current example, the rate of degradation of protein X depends on its mean lifetime τX, which takes into account active degradation as well as its dilution as the cell grows. The degradation rate depends on the concentration of protein X and is calculated by dividing this concentration by the lifetime (Figure 8–74A).


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Cell_Biology_Alberts. With equations for rates of production and degradation in hand, we can now generate a differential equation to determine the rate of change of protein X as a function of time (Equation 8–5, Figure 8–74B). This equation can be solved by the numerical methods mentioned earlier. According to the solution of this equation, when transcription begins, the concentration of protein X rises to a steady-state level at which the concentration of X is not changing anymore; that is, its rate of change is zero. When this occurs, rearrangement of Equation 8–5 yields an equation that can be used to determine the steady-state value of X, [Xst] (Equation 8–6, Figure 8–74C). An important concept emerges from the mathematics: the steady-state concentration of a gene product is directly proportional to its lifetime. If lifetime doubles, protein concentration doubles as well. The Time Required to Reach Steady State Depends on Protein Lifetime


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Cell_Biology_Alberts. We can see from Equation 8–6 (see Figure 8–74C) that when the concentration of protein A rises, protein X increases to a new steady-state value, [Xst]. But this cannot happen instantaneously. Instead, X changes dynamically according to the solution of its differential rate equation (Equation 8–5). The solution of this equation reveals that the concentration of X over time is related to its steady-state concentration according to the equation in Figure 8–74D. Once again, mathematics uncovers a simple but important concept that is not intuitively obvious: following a sudden increase in [A], [X] rises to a new steady state at an exponential rate that is inversely related to its lifetime; the faster X is degraded, the less time it takes it to reach its new steady-state value (Figure 8–74E). The faster response time comes at a higher metabolic cost, however, since proteins with a rapid response time must be produced and degraded at a high rate. For proteins that are not rapidly turned


Cell_Biology_Alberts

Cell_Biology_Alberts. The faster response time comes at a higher metabolic cost, however, since proteins with a rapid response time must be produced and degraded at a high rate. For proteins that are not rapidly turned over, the response time is very long, and protein concentration is determined primarily by the dilution that results from cell growth and division.


Cell_Biology_Alberts

Cell_Biology_Alberts. Positive control is not the only mechanism that cells use to regulate the expression of their genes. As we discussed in Chapter 7, cells also actively shut off genes, often by employing transcription repressor proteins that bind to specific sites on target genes, thereby blocking access to RNA polymerase. We can analyze the function of these repressors by the same quantitative methods described above for transcription activators. If a repressor protein R binds to the regulatory region of gene X and represses its transcription, then the fraction of gene binding sites occupied by the repressor is specified by the same equation we used earlier for the transcription activator (Figure 8–75A). In this case, however, it is only when the DNA is free that RNA polymerase can bind to the promoter and transcribe the gene. Thus, the quantity of interest is the unbound fraction, which can be viewed as the probability that the site is free, averaged over multiple binding and unbinding events. When


Cell_Biology_Alberts

Cell_Biology_Alberts. transcribe the gene. Thus, the quantity of interest is the unbound fraction, which can be viewed as the probability that the site is free, averaged over multiple binding and unbinding events. When the repressor concentration is zero, the unbound fraction is 1 and the promoter is fully active; when the repressor concentration greatly exceeds 1/K, the unbound fraction approaches zero. Figures 8–75B and C compare these relationships for a transcription activator and a transcription repressor.


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Cell_Biology_Alberts. We can create a differential equation that provides the rate of change in protein X when repressor concentrations change (Equation 8–7, Figure 8–75D). As in the case of the transcription activator, the steady-state concentration of protein X increases as its lifetime increases, but it decreases as the concentration of the transcription repressor increases. 1.0 Thus far, we have considered simple regulatory systems of just a few components. In most of the complex regulatory systems that govern cell behaviors, multiple


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Cell_Biology_Alberts. Figure 8–75 How promoter occupancy depends on the binding affinity of a transcription regulator protein. (A) The fraction of a binding site that is occupied by a transcription repressor R is determined by an equation that is similar to the one we used for a transcription activator (see Figure 8–72E), except that in the case of a repressor we are interested primarily in the unbound fraction. (B) For a transcription activator A, half of the promoters are occupied when [A] = 1/KA. Gene activity is proportional to this bound fraction. (C) For a transcription repressor R, gene activity is proportional to the unbound fraction of promoters. As indicated, this fraction is reduced to half of its maximal value when [R]=1/KR. (D) As in the case of the transcription activator A (see Figure 8–74), 0.5 0.5 we can derive equations to assess the timing of protein X production as a function of repressor concentrations.


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Cell_Biology_Alberts. modules are linked to produce larger circuits that we call network motifs, which can produce surprisingly complex and biologically useful responses whose properties become apparent only through mathematical analysis. A particularly common and important network motif is the negative feedback loop, which can have dramatically different functions depending on how it is structured.


Cell_Biology_Alberts

Cell_Biology_Alberts. We take as a first example a network motif consisting of two linked modules (Figure 8–76A). Here, an input signal initiates the transcription of gene A, which produces a transcription activator protein A. This activates gene R, which synthesizes a transcription repressor protein R. Protein R in turn binds to the promoter of gene A to inhibit its expression. This cyclical organization creates a negative feedback loop that one can intuitively understand as a mechanism to prevent proteins from accumulating to high levels. But what can we learn about negative feedback loops, and their value in biology, by using mathematics to model them?


Cell_Biology_Alberts

Cell_Biology_Alberts. The negative feedback loop in Figure 8–76A can be modeled using Equation 8–7 (see Figure 8–75D) for the repression of gene A and Equation 8–5 (see Figure 8–74B) for the activation of gene R. Thus, for proteins A and R, we use the set of differential equations (Equation set 8–8) shown in Figure 8–76B. The two equations in this set are coupled, which means that they must be solved together to describe d[A] ˜A.mA [A] = ˜R.mR – 1 0.5 0 fraction steady-stateprotein level


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Cell_Biology_Alberts. Figure 8–76 A simple negative feedback motif. (A) Gene A negatively regulates its own expression by activating gene R. The product of gene R is a transcription repressor that inhibits gene A. (B) Equation set 8–8 can be solved to determine the dynamics of system components over time. (C) A system with negative feedback (blue) reaches its steady state faster than a system with no feedback (red). The plots indicate the levels of protein A, expressed as a fraction of the steady-state level. The blue line reflects the solution of Equation set 8–8, which includes negative feedback of gene A by the repressor R. The red line represents the solution when the rate of synthesis of A was set to a constant value that is unaffected by the repressor R. (C) time the behavior of A and R over time for any value of the input. As before, we plug in values for the parameters (βR, τR, etc.) and then use a computer to determine the values of [A] and [R] as a function of time after a sudden input activates


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Cell_Biology_Alberts. value of the input. As before, we plug in values for the parameters (βR, τR, etc.) and then use a computer to determine the values of [A] and [R] as a function of time after a sudden input activates gene A.


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Cell_Biology_Alberts. The results reveal several important properties of negative feedback. First, rather surprisingly, negative feedback increases the speed of the response to the activating input. As shown in Figure 8–76C, the system with negative feedback reaches its new steady state faster than the system with no feedback. Second, negative feedback is useful for protecting cells from perturbations that continuously arise in the cell’s internal environment—due either to random variations in the birth and death of molecules or to fluctuations in environmental variables such as temperature and nutritional supplies. Let us imagine, for example, that βA, the transcription rate constant for gene A, fluctuates by 25% of its value and ask whether and how much the levels of protein R are affected. The results, shown in Figure 8–77, reveal that a change in βA causes a smaller change in the steady-state value of R when the network has negative feedback.


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Cell_Biology_Alberts. A beautiful thing happens when a negative feedback loop contains some delay mechanism that slows the feedback signal through the loop: rather than generating a new stable state as in a rapid negative feedback loop, a delayed loop generates pulses, or oscillations, in the levels of its components. This can be seen, for example, if the number of components in a negative feedback loop increases, which leads to delays in the amount of time required for the cycle of signals to be completed. Figure 8–78 compares the behavior of two network motifs—one with a three-stage and one with a five-stage negative feedback loop. Using the same kinetic parameters at each stage in the two loops, one finds that stable oscillations arise in the longer loop, while in the shorter loop the same parameters lead to relatively rapid convergence to a stable steady state.


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Cell_Biology_Alberts. Changes in the parameters of a delayed negative feedback loop—binding affinities, transcription rates, or protein stabilities, for example—can change the amplitude and period of the oscillations, providing a remarkably versatile mechanism for generating all sorts of oscillators that can be used for various purposes in the cell. Indeed, many naturally occurring oscillators, including the calcium oscillators described in Chapter 15 and the cell-cycle network described in Chapter 17, use delayed negative feedback as the basis for biologically important oscillations. Not all of the oscillations observed in cells are thought to have a function, however. Oscillations become inevitable in a highly complex, multicomponent biochemical pathway like glycolysis, due simply to the large number of feedback loops that appear to be required for its regulation.


Cell_Biology_Alberts

Cell_Biology_Alberts. We have focused thus far on the binding of a single transcription regulator to a single site in a gene promoter. Many promoters, however, contain multiple adjacent binding sites for the same transcription regulator, and it is not uncommon for these regulators to interact with each other on the DNA to form dimers or larger oligomers. These interactions can result in a cooperative form of DNA binding,


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Cell_Biology_Alberts. Figure 8–77 The effect of fluctuations in kinetic rate constants on a system with negative feedback compared to one without feedback. The plot at left represents the levels of protein R after a sudden activating stimulus, according to the regulatory scheme in Figure 8–76A and determined by the solution of Equation set 8–8 (see Figure 8–76B). A perturbation was induced by changing βA from 4 M/min (red line) to 3 M/min (blue line). The plot at right shows the results when negative feedback was removed. The system with negative feedback deviates less from its normal operation as β changes than does the system with no feedback. Notice that, as in Figure 8–76C, the system with negative feedback also reaches its steady state more rapidly.


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Cell_Biology_Alberts. such that DNA-binding affinity increases at higher concentrations of the transcription regulator. Cooperativity produces a steeper transcriptional response to increasing regulator concentration than the response that can be generated by the binding of a monomeric protein to a single site. A steep transcriptional response of this sort, when present in conjunction with positive feedback, is an important ingredient for producing systems with the ability to switch between different discrete phenotypic states. To begin to understand how this occurs, we need to modify our equations to include cooperativity.


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Cell_Biology_Alberts. Cooperative binding events can produce steep S-shaped (or sigmoidal) relationships between the concentration of regulatory protein and the amount bound on the DNA (see Figure 15–16). In this case, a number called the Hill coefficient (h) describes the degree of cooperativity, and we can include this coefficient in our equations for calculating the bound fraction of promoter (Figure 8–79A). As the Hill coefficient increases, the dependence of binding on protein concentration becomes steeper (Figure 8–79B). In principle, the Hill coefficient is similar to the number of molecules that must come together to generate a reaction. In practice, however, cooperativity is rarely complete, and the Hill coefficient does not reach Figure 8–78 Oscillations arising from delayed negative feedback.


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Cell_Biology_Alberts. Figure 8–78 Oscillations arising from delayed negative feedback. A transcriptional circuit with three components (A, B) is less likely to oscillate than a transcriptional circuit with five components (C, D). The X (light blue), Y (dark blue), and Z (brown) here represent transcription regulatory proteins. For the simulations in (B) and (D), the system was initiated from random initial conditions for X, Y, and Z. Oscillations are produced by a delay induced as the signal propagates through the loop.


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Cell_Biology_Alberts. Figure 8–79 How the cooperative binding of transcription regulatory proteins affects the fraction of promoters bound. (A) Cooperativity is incorporated into our mathematical models by including a Hill coefficient (h) in the equations used previously to determine the fraction of bound promoter (see Figures 8–72E and 8–75A). When h is 1, the equations shown here become identical to the equations used previously, and there is no cooperativity. (B) The left panel depicts a cooperatively bound transcription activator and the right panel depicts a cooperatively 0.5 bound transcription repressor. Recall from Figure 8–75B that gene activity is proportional to bound activator (left panel) or unbound repressor (right panel). 1/KR Note that the plots get steeper as the Hill (B) concentration of protein A concentration of protein R coefficient increases.


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Cell_Biology_Alberts. 1/KR Note that the plots get steeper as the Hill (B) concentration of protein A concentration of protein R coefficient increases. We turn now to positive feedback and its very important consequences. First and foremost, positive feedback can make a system bistable, enabling it to persist in either of two (or more) alternative steady states. The idea is simple and can be conveyed by drawing an analogy with a candle, which can exist either in a burning state or in an unlit state. The burning state is maintained by positive feedback: the heat generated by burning keeps the flame alight. The unlit state is maintained by the absence of this feedback signal: so long as sufficient heat has never been applied, the candle will stay unlit.


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Cell_Biology_Alberts. For the biological system, as for the candle, bistability has an important corollary: it means that the system has a memory, such that its present state depends on its history. If we start with the system in an Off state and gradually rack up the concentration of the activator protein, there will come a point where autostimulation becomes self-sustaining (the candle lights), and the system moves rapidly to an On state. If we now intervene to decrease the level of activator, there will come a point where the same thing happens in reverse, and the system moves rapidly back to an Off state. But the transition points for switching on and switching off are different, and so the current state of the system depends on the route by which it has been taken in the past—a phenomenon called hysteresis.


Cell_Biology_Alberts

Cell_Biology_Alberts. A simple case of positive feedback can be seen in a regulatory system in which a transcription regulator activates (directly or indirectly) its own expression, as in Figure 8–80A. Positive feedback can also arise in a circuit with many intervening repressors or activators, so long as the net overall effect of the interactions is activation (Figure 8–80B and C).


Cell_Biology_Alberts

Cell_Biology_Alberts. To illustrate how positive feedback can generate stable states, let us focus on a simple positive feedback loop containing two repressors, X and Y, each of which inhibits expression of the other (Figure 8–81A). As we saw with Equation set 8–8 (Figure 8–76B) earlier, we can create differential equations describing the rate of change of [X] and [Y] (Equation set 8–9, Figure 8–81B). We can further modify these equations to include cooperativity by adding Hill coefficients. As we did earlier, we can then create equations for calculating the concentrations of [X] and [Y] when the system reaches a steady state (that is, when (d[X]/dt ) = 0 and (d[Y]/dt ) = 0; Equations 8–10 and 8–11, Figure 8–81C).


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Cell_Biology_Alberts. Equations 8–10 and 8–11 can be used to carry out an intriguing mathematical procedure called a nullcline analysis. These equations define the relationships between the concentration of X at steady state, [Xst], and the concentration of Y at steady state, [Yst], which must be simultaneously satisfied. We can plug in different values for [Yst] in Equation 8–10, and calculate the corresponding [Xst] for each of these values. We can then graph [Xst] as a function of [Yst]. Next, we repeat the process by varying [Xst] in Equation 8–11 to graph the resulting [Yst]. The intersections of these two graphs determine the theoretically possible steady states of the system. For systems in which the Hill coefficients hX and hY are much larger than 1, the lines in the two graphs intersect at three locations (Figure 8–81D). In other systems that have the same arrangement of regulators but different parameters, there might only be one intersection, indicating the presence of only a single


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Cell_Biology_Alberts. Figure 8–80 Positive feedback of a gene onto itself through serially connected interactions. A sequence of activators and repressors of any length can be connected to produce a positive feedback loop, as long as the overall sign is positive. Because the negative of a negative is positive, not only circuit (A) and (B) but also circuit (C) create positive feedback. = ˜X.mX – dt 1 + (KY[Y])hY ˜X = ˜Y.mY – GENE Y dt 1 + (KX[X])hX ˜Y (B) concentration of X = ˜X.mX.˜X = ˜Y.mY.˜Y


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Cell_Biology_Alberts. Figure 8–81 A graphical nullcline analysis. (A) X inhibits Y and Y inhibits X, resulting in a positive feedback loop. (B) Equation set 8–9 can be used to determine the rate of change in the concentrations of proteins X and Y. (C) Equations 8–10 and 8–11 provide the concentrations of proteins X and Y, respectively, when these concentrations reach concentration of X a steady state. (D, E) Blue curves (called nullclines) are plots of [Xst]calculated from Equation 8–10 over a range of concentrations of [Yst]. Red curves indicate values of [Yst] calculated from Equation 8–11 over a range of concentrations of [Xst]. At an intersection of the two lines, both [X] and [Y] are at steady state. For plot (D), the binding of both proteins to their target gene promoters was cooperative (hX and hY much larger than 1), resulting in the presence of multiple intersections of the nullclines––suggesting that the system can assume multiple discrete steady states. In plot (E), the binding of protein X to


Cell_Biology_Alberts

Cell_Biology_Alberts. larger than 1), resulting in the presence of multiple intersections of the nullclines––suggesting that the system can assume multiple discrete steady states. In plot (E), the binding of protein X to the promoter of gene Y was not cooperative (hX close to 1), resulting in only one nullcline intersection and thus just one likely steady state.


Cell_Biology_Alberts

Cell_Biology_Alberts. steady state. For example, when there is a low cooperativity of protein X binding to the promoter of gene Y (that is, a small Hill coefficient, hX, in Equation 8–11), the plot of [Y] is less curved (Figure 8–81E), and it is less likely that there will be multiple intersections of the two curves.


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Cell_Biology_Alberts. We emphasized earlier that positive feedback typically generates a bistable system with two stable steady states. Why does the system modeled in Figure 8–81D have three? This conundrum can be explained by solving the reaction rate equations (Equation set 8–9, Figure 8–81B) for various different starting conditions of [X] and [Y], determining all values of [X] and [Y] as a function of time. Starting Figure 8–82 Analysis of the stability of a system’s steady states. (A) The dotted with each set of initial concentrations of [X] and [Y], these calculations produce lines are the nullclines for the system shown a so-called trajectory of points, each indicated by a curved green line on Figure in Figure 8–81. Also shown are dynamic 8–82A. A fascinating pattern emerges: each trajectory moves across the plot and trajectories (green) that show the changes settles in one of two steady states, but never in the third (middle steady state). over time in [X] and [Y], starting at a variety We conclude


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Cell_Biology_Alberts. the plot and trajectories (green) that show the changes settles in one of two steady states, but never in the third (middle steady state). over time in [X] and [Y], starting at a variety We conclude that the middle steady state is unstable because it cannot “attract” of different initial concentrations (determined by solution of Equation set 8–9; see Figure any trajectories. The system therefore has only two stable steady states. Thus, the 8–81B). By plotting [X] versus [Y] at each number of stable steady states in a system need not be equal to the total number time point, we find that, although there are of its theoretically possible steady states. In fact, stable steady states are usually three possible steady states in this system, separated by unstable ones, as in our example. the dynamic trajectories converge on only Once this system adopts a fate by settling in one of the two steady states, does two of them. The middle steady state is avoided: it is unstable, being unable to it


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Cell_Biology_Alberts. trajectories converge on only Once this system adopts a fate by settling in one of the two steady states, does two of them. The middle steady state is avoided: it is unstable, being unable to it have the ability to switch to the other state? The numerical solution of Equation attract any trajectories. (B) Imagine that the system is at the upper-left steady state and experiences a perturbation (black arrows), such as a random fluctuation in the production rates of X and/or Y. If the perturbation is small (arrow 1), the system will return to the same steady state. On the other hand, a perturbation that drives the system beyond the unstable (middle) steady state (arrow 2) causes it to switch to the lower-right steady state. The set of perturbations that a system can withstand without switching from one steady state to the other is known as the region of concentration of X concentration of X attraction of that steady state.


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Cell_Biology_Alberts. set 8–9 can again provide an answer. In Figure 8–82B, we show the solution of this equation set for two perturbations from the upper-left steady state. For a small perturbation, the system returns to its original steady state. But the larger perturbation causes the system to switch to the alternate steady state. Thus, this system can be switched from one stable steady state to the other by subjecting it to an input (or a perturbation) that is large enough to make the other steady state more attractive. More generally, every stable steady state has a corresponding region of attraction, which can be intuitively thought of as the range of perturbations (of [X] or [Y] in this example) for which the dynamic trajectories converge back to that particular steady state, rather than switch to the other one.


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Cell_Biology_Alberts. The concept of a region of attraction has interesting implications for the heritability of transcriptional states and the transition rates between them. If the region of attraction around one steady state is large, for example, then most cells in the population will assume this particular state. Furthermore, this state is likely to be inherited by daughter cells, since minor perturbations, like those ensuing from an asymmetric distribution of molecules during cell division, will rarely be sufficient to induce switching to the other steady state. We should expect that the use of positive feedback, coupled to cooperativity, will quite often be associated with systems requiring stable cell memory. Robustness Is an Important Characteristic of Biological Networks


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Cell_Biology_Alberts. Biological regulatory systems are exposed to frequent and sometimes extreme variations in external conditions or the concentrations or activities of key components. The ability of these systems to function normally in the face of such perturbations is called robustness. If we understand a complex system to the extent that we can reproduce its behavior with a computational model, then the robustness of the system can be assessed by determining how well its normal function persists following changes in various parameters, such as rate constants and component concentrations. We have already seen, for example, how the presence of negative feedback reduces the sensitivity of the steady state to changes in the values of the system’s parameters (see Figure 8–77). Considerations of robustness also apply to dynamic behaviors. Thus, for example, when discussing negative feedback, we described how the behavior of a system tends to become more oscillatory as the number of components that


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Cell_Biology_Alberts. also apply to dynamic behaviors. Thus, for example, when discussing negative feedback, we described how the behavior of a system tends to become more oscillatory as the number of components that constitute the feedback loop increases. If we use different values of the parameters in models derived for systems like those in Figure 8–78, we find that the system with the longer loop tends to exhibit stable oscillations within a much broader range of parameters, indicating that this system provides a more robust oscillator. We can perform similar calculations to determine the ability of different systems to achieve robust bistability arising from positive feedback. Thus, one benefit of computational models is that they allow us to probe the robustness of biological networks in a systematic and rigorous way.


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Cell_Biology_Alberts. Two Transcription Regulators That Bind to the Same Gene Promoter Can Exert Combinatorial Control Thus far, we have discussed how one transcription regulator can modulate the expression level of a gene. Most genes, however, are controlled by more than one type of transcription regulator, providing combinatorial control that allows two or more inputs to influence the expression of one gene. We can use computational methods to unveil some of the important regulatory features of combinatorial control systems.


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Cell_Biology_Alberts. Consider a gene whose promoter contains binding sites for two regulatory proteins, A and R, which bind to their individual sites independently. There are four possible binding configurations (Figure 8–83A). Suppose that A is a transcription activator, R is a transcription repressor, and the gene is only active when A is bound and R is not bound. We learned earlier that the probability that A is bound and the probability that R is not bound can be determined by the equations in Figure 8–84A. The product of these two probabilities gives us the probability of gene activation.


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Cell_Biology_Alberts. This example illustrates an AND NOT logic function (A and not R) (see Figure 8–83A). Maximal activation of this gene is accomplished when [A] is high and [R] is zero. However, intermediate levels of gene activation are also possible depending on the levels of A and R and also on the binding affinities of [A] and [R] for their respective sites (that is, KA and KR). When KA » KR, even a small concentration of [A] is capable of overcoming repression by R. Conversely, if KA « KR, then much more [A] is needed to activate the gene (Figure 8–84B and C).


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Cell_Biology_Alberts. Many other logic functions can govern combinatorial gene regulation. For example, an AND logic gate results when two activators, A1 and A2, are both required for a gene to be transcribed (Figures 8–83B and 8–84D). In E. coli cells, the AraJ gene controls some aspects of arabinose sugar metabolism: its expression requires two transcription regulators, one activated by arabinose and the other activated by the small molecule cAMP (Figure 8–84E). KA[A]fraction of A bound = fraction of R not bound = P(A,R) = . = concentration of R concentration of R concentration of A concentration of A


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Cell_Biology_Alberts. KA[A]fraction of A bound = fraction of R not bound = P(A,R) = . = concentration of R concentration of R concentration of A concentration of A Figure 8–83 Combinatorial control of gene expression. There are many ways in which gene expression can be controlled by two transcription regulators. To define precisely the relationship between the two inputs and the gene expression output, a regulatory circuit is often described as a specific type of logic gate, a term borrowed from electronic circuit design. A simple example is the OR logic gate (not shown here), in which a gene is controlled by two transcription activators, and one or the other can activate gene expression. (A) In a system with an activator A and repressor R, if transcription is turned on only when A is bound and R is not, then the result is an AND NOT logic gate. We saw an example of this logic in Chapter 7 (Figure 7–15).


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Cell_Biology_Alberts. (B) An AND gate results when two transcription activators, A1 and A2, are both required to turn on a gene. Figure 8–84 How the quantitative output of a gene depends on both its combinatorial logic and the affinities of transcription regulators. (A) In a combinatorial gene regulatory system like that illustrated in Figure 8–83A, the fraction of promoters bound by activator A and not bound by repressor R are each determined as shown here. The product of these probabilities provides the probability, P(A, R), that a gene promoter is active. (B–E) In these four panels, red indicates high gene expression and blue indicates low gene expression. (B) and (C) depict gene expression from the system described in panel (A). The two panels demonstrate how the system behaves when the relative concentration of A1 concentration of A2 arabinose (mM) S. Kaplan et al., Mol. Cell 29:786–792, 2008.) 0.02 1.3 43 affinities of the two transcription regulators change as indicated above each panel.


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Cell_Biology_Alberts. (D) Gene expression in a case where the gene turns on only at high levels of both activating inputs (A1 and A2), as shown in Figure 8–83B. (E) Experimental data showing measured expression of a gene in E. coli that is combinatorially regulated by two inputs: arabinose and cAMP. Note the close resemblance to panel (D). (E, adapted from rate of protein synthesis rate of protein synthesis


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Cell_Biology_Alberts. Imagine that a sudden input signal immediately activates a transcription activator A and that the same input signal induces the much slower synthesis of a transcription repressor protein R that acts on the same gene X. If A and R control gene expression by an AND NOT logic function like that described above, our intuition tells us that this system should be able to generate a pulse of transcription: when A is activated (and R is absent), the transcription of gene X will begin and cause an increase in the concentration of protein X, but then transcription will shut off when the concentration of R increases to a sufficiently high value.


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Cell_Biology_Alberts. Arrangements of this type are common in the cell. In E. coli, for example, galactose metabolic genes are positively regulated by the catabolite activator protein (CAP), which is activated at high levels of cAMP. The same genes are repressed by the GalS repressor protein, which is encoded by a gene whose transcription is likewise activated by CAP. Thus, an increase in input (cAMP) activates A (CAP), and transcription of the galactose genes begins. But activation of A also causes a subsequent buildup of R (GalS), which causes the same genes to be repressed after a delay. This results in an incoherent feed-forward motif (Figure 8–85A).


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Cell_Biology_Alberts. The response of the incoherent feed-forward motif will vary, depending on the parameters of the system. Suppose, for example, that the transcription activator protein A binds more weakly to the gene regulatory region than does the transcription repressor protein R (KA < KR). In this case, there will be a transient burst of protein synthesized by the affected gene (gene X) in response to a sudden activating input (Figure 8–85B). In contrast, the output will be more sustained if KA is much larger than KR, because the repression will be too weak to overcome the gene activation (Figure 8–85C). Other properties of this network, such as the dependence of the amplitude of the pulse on the various rate constants in the system, can be explored with the same computational tools. Thus, our intuitive guess about how this system would behave was only partially correct; even the simplest of networks depends on precise interaction strengths, demonstrating yet again why mathematics is needed to


Cell_Biology_Alberts

Cell_Biology_Alberts. guess about how this system would behave was only partially correct; even the simplest of networks depends on precise interaction strengths, demonstrating yet again why mathematics is needed to complement cartoon drawings.


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Cell_Biology_Alberts. In the bacterium E. coli, the sugar arabinose is only consumed when the preferred sugar, glucose, is scarce. The strategy that cells use to assess the presence of arabinose and absence of glucose involves a feed-forward arrangement that is different from the one just described. In this case, depletion of glucose causes an increase of cAMP, which is sensed by the CAP transcription activator protein, as described previously. In this case, however, CAP also induces the synthesis of a second transcription activator, AraC. Both activator proteins are necessary to activate arabinose metabolic genes (the AND logic function in Figure 8–83B). Figure 8–85 How an incoherent feed-forward motif can generate a brief pulse of gene activation in response to a sustained input. (A) Diagram of an incoherent feed-forward motif in which the transcription activator A and the repressor R control the expression of gene X using the AND NOT logic of Figure 8–83A.


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Cell_Biology_Alberts. (B) When KA « KR, this motif generates a pulse of protein X expression, such that the output goes back down even if the input remains high. (C) When KA » KR, the same motif responds to a sustained input by generating a sustained output.


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Cell_Biology_Alberts. This arrangement, known as a coherent feed-forward motif, has the interesting characteristics illustrated in Figure 8–86. Imagine that two activators, A1 and A2, are both required to initiate transcription of a gene. The input to the network activates A1 directly, but only activates A2 through this A1 activation. Thus, for a protein to be synthesized from this gene, long-term inputs are required that allow both A1 and A2 to be produced in active form. Brief input pulses are either ignored or produce small outputs. The requirement for a long input is important if assurances about a signal are needed before a costly cellular program is triggered. For example, glucose is the sugar on which E. coli cells grow best. Before cells trigger arabinose metabolism in the example above, it might be beneficial to be sure that glucose has been depleted (a sustained CAP pulse), rather than inducing the arabinose program during a transient glucose fluctuation.


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Cell_Biology_Alberts. The Same Network Can Behave Differently in Different Cells Due to Stochastic Effects Up to this point, we have assumed that all cells in a population produce identical behaviors if they contain the same network. It is important, however, to account for the fact that cells often show considerable individuality in their responses. Consider a situation in which a single mother cell divides into two daughter cells of equal volume. If the mother cell has only one molecule of a given protein, then only one daughter will inherit it. The daughters, though genetically identical, are already different. This variability is most pronounced for molecules that are present in small numbers. Nevertheless, even when there are many copies of a particular protein (or RNA), it is very unlikely that both daughter cells will end up with exactly the same number of molecules.


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Cell_Biology_Alberts. This is just one illustration of a universal feature of cells: their behaviors are often stochastic, meaning that they display variability in their protein content and therefore exhibit variations in phenotypes. In addition to the asymmetric partitioning of molecules following cell division, variability can originate from many chemical reactions. Imagine, for example, that our mother cell contains a simple gene regulatory circuit with a positive feedback loop like that shown in Figure 8–80B. Even if both daughter cells receive a copy of this circuit, including one copy of the initial transcription activator protein, there will be variability in the time required for promoter binding—and it will be statistically nearly impossible for the genes in the two daughter cells to become activated at precisely the same time. If the system is bistable and poised near a switching point, then variability in the response might flip the switch in only one daughter cell. Two daughter cells that were


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Cell_Biology_Alberts. at precisely the same time. If the system is bistable and poised near a switching point, then variability in the response might flip the switch in only one daughter cell. Two daughter cells that were born identical can thereby acquire, by chance, a dramatic difference in phenotype.


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Cell_Biology_Alberts. More generally, isogenic populations of cells grown in the same environment display diversity in size, shape, cell-cycle position, and gene expression. These differences arise because biochemical reactions require probabilistic collisions Figure 8–86 How a coherent feed-forward motif responds to various inputs. (A) Diagram of a coherent feed-forward motif in which the transcription activators A1 and A2 together activate expression of gene X using the AND logic of Figure 8–83B. (B) The response to a brief input can be either weak (as shown) or nonexistent. This allows the motif to ignore random fluctuations in the concentration of signaling molecules. (C) A prolonged input produces a strong response that can turn off rapidly.


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Cell_Biology_Alberts. between randomly moving molecules, with each event resulting in changes in the number of molecular species by integer amounts. The amplified effect of fluctuations in a molecular reactant, or the compounded effects of fluctuations across many molecular reactants, often accumulates as an observable phenotype. This can endow a cell with individuality and generate non-genetic cell-to-cell variability in a population.


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Cell_Biology_Alberts. Non-genetic variability can be studied in the laboratory by single-cell measurements of fluorescent proteins expressed from genes under the control of a specific promoter. Live cells can be mounted on a slide and viewed through a fluorescence microscope, revealing the striking variability in protein expression levels (Figure 8–87). Another approach is to use flow cytometry, which works by streaming a dilute suspension of cells past an illuminator and measuring the fluorescence of individual cells as they flow past the detector (see Figure 8–2). Fluorescence values can be used to build histograms that reveal the variability in a process across a population of cells, with a broad histogram indicating higher variability. Several Computational Approaches Can Be Used to Model the Reactions in Cells


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Cell_Biology_Alberts. We have focused primarily on the use of ordinary differential equations to model the dynamics of simple regulatory circuits. These models are called deterministic, because they do not incorporate stochastic variability and will always produce the same result from a specific set of parameters. As we have seen, such models can provide useful insights, particularly in the detailed mechanistic analysis of small regulatory circuits. However, other types of computational approaches are also needed to comprehend the great complexity of cell behavior. Stochastic models, for example, attempt to account for the very important problem of random variability in molecular networks. These models do not provide deterministic predictions about the behavior of molecules; instead, they incorporate random variation into molecule numbers and interactions, and the purpose of these models is to obtain a better understanding of the probability that a system will exist in a certain state over time.


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Cell_Biology_Alberts. Numerous other modeling strategies have been or are being developed. Boolean networks are used for the qualitative analysis of complex gene regulatory networks containing large numbers of interacting components. In these models, each molecule is a node that can exist in either the active or inactive state, thereby affecting the state of the nodes it is linked to. Models of this sort provide insights into the flow of information through a network, and they were useful in helping us understand the complex gene regulatory network that controls the early development of the sea urchin (see Figure 7–43). Boolean networks therefore reduce complex networks to a highly simplified (and potentially inaccurate) form. At the other extreme are agent-based simulations, in which thousands of molecules (or “agents”) in a system are modeled individually, and their probable behaviors and interactions with each other over time are calculated on the basis of predicted physical and chemical behaviors,


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Cell_Biology_Alberts. (or “agents”) in a system are modeled individually, and their probable behaviors and interactions with each other over time are calculated on the basis of predicted physical and chemical behaviors, often while taking stochastic variation into account. Agent-based approaches are computationally demanding but have the potential to generate highly lifelike simulations of real biological systems.


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Cell_Biology_Alberts. Statistical Methods Are Critical For the Analysis of Biological Data Dynamics, differential equations, and theoretical modeling are not the be-all and end-all of mathematics. Other branches of the subject are no less important for biologists. Statistics—the mathematics of probabilistic processes and noisy datasets—is an inescapable part of every biologist’s life. This is true in two main ways. First, imperfect measurement devices and other errors generate experimental noise in our data. Second, all cell-biological processes depend on the stochastic behavior of individual molecules, as we just discussed, and this results in biological noise in our results. How, in the face of all this noise, do we come to conclusions about the truth of hypotheses? The answer is statistical analysis, which shows how to move from one level of description to


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Cell_Biology_Alberts. Figure 8–87 Different levels of gene expression in individual cells within a population of E. coli bacteria. For this experiment, two different reporter proteins (one fluorescing green, the other red), controlled by a copy of the same promoter, have been introduced into all of the bacteria. Some cells express only one gene copy, and so appear either red or green, while others express both gene copies, and so appear yellow. This experiment reveals variable levels of fluorescence, indicating variable levels of gene expression within an apparently uniform population of cells. (From M.B. Elowitz et al., Science 297:1183–1186, 2002. With permission from AAAS.) another: from a set of erratic individual data points to a simpler description of the WHAT WE DON’T KNOW key features of the data.


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Cell_Biology_Alberts. Statistics teaches us that the more times we repeat our measurements, the bet • Many of the tools that revolutionized ter and more refined the conclusions we can draw from them. Given many repeti-DNA technology were discovered by tions, it becomes possible to describe our data in terms of variables that summa-scientists studying basic biological rize the features that matter: the mean value of the measured variable, taken over problems that had no obvious the set of data points; the magnitude of the noise (the standard deviation of the applications. What are the best set of data points); the likely error in our estimate of the mean value (the standard strategies to ensure that such crucially important technologies will continue to error of the mean); and, for specialists, the details of the probability distribution be discovered?


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Cell_Biology_Alberts. describing the likelihood that an individual measurement will yield a given value. For all these things, statistics provides recipes and quantitative formulas that biol • As the cost of DNA sequencing ogists must understand if they are to make rigorous conclusions on the basis of decreases and the amount of variable results.


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Cell_Biology_Alberts. sequence data accumulates, how are we going to keep track of and Summary meaningfully analyze this vast amount of information? What new questions Quantitative mathematical analysis can provide a powerful extra dimension in our will this information allow us to answer? understanding of cell regulation and function. Cell regulatory systems often depend • Can we develop tools to analyze on macromolecular interactions, and mathematical analysis of the dynamics of each of the post-transcriptional these interactions can unveil important insights into the importance of binding modifications on the proteins in living affinities and protein stability in the generation of transcriptional or other signals. cells, so as to follow all of their Regulatory systems often employ network motifs that generate useful behaviors: changes in real time?