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More concepts from ring theory

Friday, February 29, 2008

We are at an important juncture in our discussion of algebraic number theory. From here on out, the path starts to go uphill more steeply, with quite a bit more abstraction and technical complexity. I hope you'll follow along anyhow. Don't feel like you need to grasp all the fine points immediately.

We're now going to cover some concepts of ring theory that are essential for talking about rings of algebraic integers. We introduced algebraic integers themselves here. The previous discussion of rings is here. Mathematics being what it is, you sort of need to have some exposure to these preliminaries in order to go further. All preceding installments in this series are listed here.

In the previous three installments we spent of lot of time on the concept of "unique factorization". What we are about to do is formalize the concept in terms of ideals of commutative rings. This discussion needs to be somewhat lengthy, but at the end we will be able to state some general properties that rings of algebraic integers have, and some conditions that are equivalent to uniqueness of factorization. Don't get too stressed by this. Detailed proofs won't be given. The level of difficulty here is comparable to what's in an introductory college course in abstract algebra or linear algebra.

Because of the importance of unique factorization, an integral domain which has unique factorization is called a unique factorization domain, or UFD. A little more precisely, an integral domain R is a UFD if there is a set P of irreducible elements such that every nonzero element α of R can be written in a unique way as a finite product α=u∏1≤k≤n pkek, where u is a unit, n is a nonnegative integer, pk are distinct elements of P, and exponents ek are nonnegative integers (all of which depend on α).

The defining property of a UFD is pretty clear and understandable, but it is not expressed in the language of ideals. As we shall see, a great deal of what we want to know about algebraic numbers can be expressed in terms of ideals, so we'd like to know how unique factorization fits in. To this end, we have this centrally important concept: An integral domain R is called a principal ideal domain (PID for short) in case every ideal I⊆R is equal to an ideal of the form (α)=αR for some α∈R.

It is a fact, which is not difficult to prove, but not immediately obvious either, that every principal ideal domain is a unique factorization domain. So if we want to show that a given integral domain R has unique factorization, it is sufficient to show that R is a PID. Unfortunately, among all integral domains there are some which are UFDs but not PIDs. So R can be a UFD even if it is not a PID – being a PID is not a necessary condition. The class of UFDs contains the class of PIDs, but is strictly larger. For instance, if F is a field, the ring of polynomials in one variable F[x] is a PID (and a UFD). (The reason this is true will be mentioned in a moment.) However, the ring of polynomials in two variables F[x,y] is a UFD but not a PID.

Obviously, it would be convenient to have a simple criterion to determine whether a domain R is a PID (and hence a UFD). It turns out that the the process of division-with-remainder that can be performed in ℤ and in polynomial rings in one variable fills the bill. All that's needed is an integer valued function on the domain R with certain properties. For a∈R, let this function be written as |a| (since it is much like an absolute value). This function should have three properties:
  1. |a|≥0 for all a∈R, and |a|=0 if and only if a=0
  2. |ab| = |a|⋅|b| for all a,b∈R
  3. if a,b∈R and b≠0, then there exist q,r∈R such that a=qb+r, with 0≤|r|<|b|
A domain R with such a function is called a Euclidean domain, because one has a Euclidean algorithm that works just as it does in ℤ. If I⊆R is a nonzero ideal, it has an element b∈I, with b≠0, of smallest nonzero "norm" |b|. If a∈I we can write a=qb+r for q,r∈R, and |r|<|b|. Yet r=a-qb is in I, so by the assumption of minimality of |b| we have |r|=0 and therefore r=0. Hence a=qb, I⊆(b), and finally I=(b) is principal. In other words, every Euclidean domain is a PID, and therefore a UFD.

Unfortunately, even among rings of integers of quadratic fields, only finitely many are known to be Eucldean. If F=ℚ(√d) and OF is the ring of integers, it is known that for d<0 the ring is Euclidean only when d=-1, -2, -3, -7, or -11. If d>0, the number of rings which are Euclidean is larger. What is known is that only a finite number of these are Euclidean using the norm function. At least one other is Euclidean using a function other than the norm function, but so far it's not known whether there are only a finite number like that.

This may seem rather disappointing, but in fact the quadratic fields where OF is a PID are also quite scarce. If d<0, then in addition to the Euclidean cases, the only other values are d=-19, -43, -67, and -163. The proof that this is a complete list for d<0 is quite difficult and was not satisfactorily done until 1966.

The situation with d>0 is even more difficult. It is not actually known whether there are only finitely many d>0 such that Oℚ(√d) is a PID. Gauss himself conjectured that there are infinitely many, but this is still an important open question.

Returning to concepts, we recall that among all integral domains, the class of PIDs is strictly smaller than the class of UFDs. It turns out that there is a subclass of all integral domains in which the notions of UFD and PID are equivalent. In fact, this is an important class, because it includes all rings of algebraic integers. This class itself can be defined by a number of equivalent conditions. But to explain this, we need to discuss the group of fractional ideals of an integral domain.

Fractional ideals

If A and B are ideals of any commutative ring R, it's easy to define the product of two ideals as a set of finite sums: A⋅B = {∑1≤k≤n akbk | ak∈A, bk∈B, n∈ℤ, n>0}. By definition of an ideal, A⋅B⊆A and A⋅B⊆B, hence A⋅B⊆A∩B. If R has a unit (as we usually assume), then clearly A⋅R=A. So in the set of ideals of R there is a commutative binary operation of multiplication, and it has an identity. Multiplication of ideals is associative since R multiplication in R is associative. (A set with an associative multiplication is called a semigroup, and if an identity exists, it's a monoid. In neither case is multiplication necessarily commutative.)

In a situation like this, it's natural (for a mathematician anyhow) to wonder what conditions on R would make the set of its ideals into a full group -- that is, how the inverse of an ideal might be defined. It turns out that for integral domains the conditions are beautiful and everything one could hope for.

To get an idea of where to start, consider the principal ideal domain ℤ. For any nonzero n∈ℤ, the obvious thing to consider is (1/n) = {m/n | m∈ℤ}. That's sort of like an ideal, since it's a commutative group under addition, and m(1/n)⊆(1/n) for all m∈ℤ. Also, under the obvious definition, (n)(1/n) = ℤ (since the product contains 1). So (1/n) surely acts like the "inverse" of the ideal (n) of ℤ for any n.

Suppose R is any commutative ring with an identity and M is any set at all (not necessarily related directly to R) where one can define an operation of multiplication rm=mr for r∈R and m∈M. Suppose further that:
  1. M is a commutative group under addition.
  2. rm∈M for all r∈R and m∈M.
  3. 1m = m for all m∈M.
  4. r(m1 + m2) = rm1 + rm2 for all r∈R, m1,m2∈M.
Then M is said to be an R-module. (If R isn't commutative, one can define R-modules by being a little more picky about the definition.) So from the example above, (1/n) is a ℤ-module, and also any ideal of a ring R is an R-module.

Given all that, if R is an integral domain, whatever a fractional ideal of R might be, it certainly should be an R-module. Indeed, we can formally define a fractional ideal of R as an R-module M such that:
  1. M⊆F, if F is the field of quotients of R.
  2. The multiplication rm for r∈R and m∈M is just the normal multiplication in F.
  3. There is some nonzero r∈R such that rM⊆R.
(As a reminder, the field of quotients of an integral domain R is defined as follows. Consider the set of pairs (m,n), with m,n∈R, n≠0 Consider two pairs (m,n) and (m′,n′) to be equivalent just in case mn′=m′n. (Think of this as fractions, where m/n = m′/n&prime when mn′=m′n.) The underlying set of the field of quotients is the set of equivalence classes of pairs under this relation. Define addition on this set by (m,n)+(m′,n′) = (mn′+m′n,nn′) and multiplication by (m,n)(m′,n′) = (mm′,nn′). Then it can be shown that addition and multiplication are well-defined, and the set of equivalence classes of pairs is indeed a field.)

Multiplication of fractional ideals is defined just like multiplication of ideals: M⋅M′= {∑1≤k≤n mkm′k | mk∈M, m′k∈M′, n∈ℤ, n>0}. With this definition, the set of fractional ideals of R is a monoid. The question is: what conditions on R will guarantee that fractional ideals form a group? This is not just a matter of idle curiosity, because it turns out that for rings with the right properties, one has unique factorization in the group of fractional ideals, and in the set of integral (i. e. ordinary) ideals as well. For rings of algebraic integers, which just happen to have the right properties, this unique factorization of ideals is almost as good, for many purposes, as having unique factorization of ring elements themselves.

We need just a few more concepts before we can state the necessary conditions. A proper ideal of R is an ideal I that is not equal to R, i. e. a proper subset. One writes I⊂R. A prime ideal P is a proper ideal such that for all a,b∈R, ab∈P only if either a∈P or b∈P (or both). In ℤ, for example, (6) isn't a prime ideal, since 2⋅3∈(6), but 2∉(6) and 3∉(6). A maximal ideal is a proper ideal P that is not properly contained in some other proper ideal P′. The integral domain R, with field of fractions F, is said to be integrally closed if every α∈F that is integral (i. e. an algebraic integer of F) over R is actually an element of R. For example, if R=ℤ, then F=ℚ, and to say α∈R is integral over F means f(α)=0 for some monic polynomial f(x) with coefficients in R, i. e. f(x)∈ℤ[x]. If α=a/b for a,b∈ℤ, then when you "clear fractions" in f(a/b)=0, you find b|a. This argument applies in any UFD, so in fact any UFD is integrally closed.

Lastly, we need a type of finiteness condition. An ideal I is said to be finitely generated if it has the form I = {∑x∈S axx | ax∈R for all x∈S}, where S⊆R is a finte set of generators. It is much like a principal ideal, except for having n=#(S) generators instead of 1. If all ideals of a ring are finitely generated, the ring is said to be Noetherian, after Emmy Noether (1882-1935). There are several equivalent characterizations of Noetherian rings. For instance, R is Noetherian if and only if every nonempty family of ideals of R has a maximal element (which contains all the other members of the family) with respect to inclusion.

Finally we can state the crucial result: If R is an integral domain, then the following are equivalent:
  1. R is Noetherian, integrally closed, and every nonzero prime ideal is maximal.
  2. Every nonzero ideal of R is uniquely expressible as a product of prime ideals.
  3. Every nonzero ideal of R is a product of prime ideals.
  4. The set of nonzero fractional ideals of R forms a group under multiplication.
The first item on this list characterizes R in terms of several ring-theoretic properties. The second item is unique factorization into prime ideals, and it is in fact equivalent to the apparently weaker third item. The fourth item is the key fact, which answers our earlier quesion about when the fractional ideals of R form a group.

Any integral domain R that has one of these properties has all of them, and is called a Dedekind domain, after Richard Dedekind (1831-1916). It isn't hard to show that if F⊇ℚ is a finite field extension, then the ring OF of algebraic integers of F has the properties listed in the first item, and so OF is a Dedekind domain and has the other properties also.

As rings, Dedekind domains have some very nice properties. For example, if R is a Dedekind domain:
  1. P is a prime ideal of R if and only if it is indecomposable, i. e. P ≠ I⋅I′ where I and I′ are ideals other than P or R.
  2. If P is a prime ideal and P divides the product I⋅I′ of two ideals (P|I⋅I′), then P|I or P|I′.
  3. Divisibility between fractional ideals is equivalent to inclusion, i. e. if M and M′ are nonzero fractional ideals, then M divides M′ if and only if M⊇M′. (Multiplication of ideals yields a result that is smaller.)
  4. If R is a unique factorization domain, it is a principal ideal domain.
  5. Every fractional ideal M of R can be generated by at most two elements, and one of these elements of M can be chosen arbitrarily.

Dedekind, Emmy Noether, and a few others were the main developers of ideal theory in this form, and Dedekind was a leading figure in the theory of algebraic numbers in general. Ernst Kummer (1810-1893) somewhat earlier had a more primitive theory of "ideal numbers" which provided a kind of unique factorization of algebraic numbers, but Dedekind made the theory much simpler and more general.

In the next installment we'll look at simple examples of how ideals that are prime in a ring of integers may split into factors in the ring of integers of an extension field. This is a very key issue in the overall theory. Eventually we will see how this abstract point of view generalizes some important ideas, called "reciprocity laws" from classical number theory.

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Spitzer Catches Young Stars in Their Baby Blanket of Dust

Thursday, February 21, 2008

Spitzer Catches Young Stars in Their Baby Blanket of Dust (2/11/08)



Rho Ophiuchi – click for 900×720 image

More: here, here

Why do we care about unique factorization?

I know there's a keen expectancy out there for the next installment of our series on algebraic number theory – so here it is. Check here for preceding installments.

In the article before the most recent one, we reviewed the fact that there is unique factorization into primes in the ring ℤ of ordinary integers. And in the most recent article, we saw that some rings of algebraic integers do not have unique factorization.

But we still haven't explained why one might care about unique factorization. That's what we're going to take up now.

One may reasonably ask why this matters so much that we've spent so much time on it. The answer is that even leading mathematicians did not clearly appreciate, until the 1840s, how unique factorization could fail in rings of algebraic integers – and they sometimes based erroneous or incomplete proofs on the assumption of unique factorization.

The first noteworthy case of this seems to have been Leonhard Euler (1707-83). Number theory was one of a vast number of Euler's interests, and one of the problems he dealt with was what is now known as Fermat's Last Theorem: The equation xn+yn=zn has no nontrivial integer solutions for integers n>2. Pierre de Fermat (1601-65) himself publicly stated this result only in case n=3 or 4. Fermat stated the case for general n only in a private note in his copy of Diophantus' Arithmetica.

No written proof of this by Fermat in even the simplest cases is known, but it can easily be proven if n=4 by a technique Fermat invented, called the "method of infinite descent". This method involves assuming a solution exists for some triple (x,y,z) and then showing that there must always be another non-trivial solution (x′,y′,z′) in which at least one of the numbers is strictly smaller than any number of the original solution. Since that's impossible, the contradiction shows there couldn't have been any solution to begin with.

Anyhow, Euler knew the proof when n=4, and set out to give a proof along the same lines when n=3. He had the brilliant and original idea to work with numbers of the form a+b√-3 for a,b∈ℤ, that is, numbers in ℤ[√-3]. This was a great idea because it provided an entirely new and powerful set of tools for dealing with questions about ordinary integers. Unfortunately, as prescient as Euler was, there were too many subtleties in this area to use the tools correctly from the start.

One step of the argument involved reasoning that if for some c∈ℤ, c3 = a2+3b2 = (a+b√-3)(a-b√-3), and if the factors a+b√-3 and a-b√-3 are relatively prime, then each of the factors must themselves be cubes of numbers in ℤ[√-3]. One problem is that ℤ[√-3] isn't the full ring of integers of ℚ(√-3), since -3≡1 (mod 4). But even disregarding that, the conclusion depends on unique factorization of the numbers involved. Although it happens to be true that unique factorization holds in the integers of ℚ(√-3), Euler didn't seem to recognize the need to prove that.

This same lack of clarity about unique factorization in rings of algebraic integers seems to have persisted into the 1840s. In 1847 Gabriel Lamé (1795-1870) thought he had a proof of Fermat's theorem for arbitrary n. Lamé worked with numbers of the form ℤ[ζn] where ζn is a root of xn-1=0 – which is called an nth root of unity. (Here we assume n is the smallest integer for which the chosen ζn satisfies the equation.)

ℤ[ζn] is called the ring of cyclotomic integers (for a particular choice of n), and it is in fact the ring of integers of the field ℚ(ζn), the nth cyclotomic field – of which we shall have much to say later on. By 1847 some astute mathematicians did recognize the need for proof of unique factorization, and they pointed it out to Lamé. He must have quickly appreciated the problem, since he didn't persist in developing his "proof".

The purported proof went something like this: Suppose there were a solution of xn+yn=zn for some n>2 and integers x, y, and z that are relatively prime. The equation could be rewritten as
xn = zn - yn = ∏1≤k≤n (z - ζnky)
Since x∈ℤ but most of the factors on the right hand side aren't, there would be a clear violation of unique factorization. Unfortunately, such a violation can't be ruled out, so the proof doesn't work. (It does work for those n where ℚ(ζn) has unique factorization. It wasn't known until 1976 that there are only 29 distinct cyclotomic fields that do have unique factorization.)

Interestingly enough, at almost exactly the same time, Eduard Kummer (1810-93), working independently on questions involving cyclotomic fields, had not only understood the problem of (lack of) unique factorization, but had even started to develop a way around the problem – what he called the method of "ideal numbers" or "divisors". Kummer had also found examples where unique factorization failed in ℤ[ζ37]. He wrote a letter to the mathematicians in Paris who were debating Lamé's work, and pretty much put an end to their deliberations.

Although Kummer's work was not solely concerned with Fermat's Last Theorem, he made what were some of the most significant partial solutions to the problem, and in the process played a huge role in advancing algebraic number theory. His work also led to the theory of ideals as discussed here. Much of what Kummer tried to do was to find "ideal" numbers, of some sort, for which unique factorization could be proven, so that as above a contradiction would arise if Fermat's equation had a solution.

In upcoming installments we'll work with somewhat more abstract ring theory, and eventually find that in any ring of algebraic integers (or in certain rings that are defined more abstractly), there is unique factorization of ideals into prime ideals.

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Wnt signaling

Tuesday, February 19, 2008

We've discussed Wnt signaling a couple of times before, here, and here.

Wnt refers to a family of proteins now numbering perhaps 20 or more, which have been found in a wide range of multicellular animals, from fruit flies, to fish, to mice and humans. Wnt proteins carry messages between cells, and are especially important in embryogenesis. They are known to play a large role in the control of stem cells and regeneration of body parts (in species where this occurs). In mammals, including humans, Wnt signaling, when it malfunctions, also seems to be involved in many types of cancer, degenerative diseases of aging, and other aging-related problems such as insulin resistance. It may be possible to ameliorate a number of these disease conditions once we have a better understanding of the details of Wnt signaling.

The "Wnt signaling pathway" refers to a sequence of proteins that, in the presence of earlier members of the pathway, change in behavior to affect proteins later in the pathway. The pathway begin when a Wnt protein (secreted by a nearby cell) binds to a cell surface protein, such as the whimsically-named Frizzled. Various other proteins in the pathway then interact, and eventually result in the build-up of a protein called β-catenin, which enters the cell nucleus, where it combines with various transcription factors to affect gene expression.

The name "Wnt" originates from the realization that two genes discovered earlier were homologous – the "wingless" gene in fruit flies (which, when mutated, yields flies without wings), and the Int genes found in mouse tumors.

Although Wnt genes and proteins have now been studied for nearly 20 years, the pace of discovery continues to increase. This is because of the large number of very interesting processes heavily influenced by Wnt signaling – from proliferation and differentiation of stem cells to embryonic development, regeneration of body parts, cancer, and degenerative diseases of aging.

The following summaries of research reports from just the past half year or so will give a buffet-style sample of Wnt-related investigations.

Carbohydrate Regulates Stem Cell Potency (2/1/08)
Embryonic stem cells are characterized by an ability to continually self-renew, but also to give rise to any adult cell type. Stem cell renewal is driven by several external signaling proteins and growth factors, including Wnt, FGF (fibroblast growth factor), and BMP (bone morphogenetic protein). In particular, Wnt signaling stimulates β-catenin to produce the transcription factor Nanog, which maintains pluripotency. However, the ability of these proteins to attach to stem cell surface proteins in order to induce a response seems to depend on the presence of a carbohydrate molecule called heparan sulfate (HS). Stem cells were found to reproduce less frequently but differentiate more frequently in proportion to experimental inhibition of HS production.

Beta-catenin Gradient Linked To Process Of Somite Formation (12/27/07)
In a developing vertebrate embryo somites are masses of a type of tissue (mesoderm) that will eventually develop into such adult tissue types as skeletal muscle and vertebrae. This research on mouse embryos demonstrates the importance of β-catenin as the principal mediator of the Wnt-signaling pathway, in the process of somite formation. In particular, there is a gradient in levels of β-catenin found in cells of the presomitic mesoderm (PSM), and this gradient is critical in regulating mesoderm maturation. This leads to the development of the characteristic vertebral column in embryos of vertebrate animals.

Certain Diseases, Birth Defects May Be Linked To Failure Of Protein Recycling System (12/20/07)
The Wnt signaling protein, like other proteins, is produced in the nuclei of certain cells, and it must be transported to the cell surface, so it can be secreted into the extracellular environment to regulate the growth of tissues during (and after) embryonic development. Another protein, called Wntless (Wls), acts as a cargo container for Wnt, and plays a key role in the transport process. Another protein, called Vps35, which makes up an important part of the "retromer complex", is responsible for moving empty Wls molecules (like freight cars) to where they are needed in the cell. But mutated Vps35 proteins can fail to perform their function, and consequently lead to the failure to transport Wnt out of the cell where it has been produced.

Grape Powder Blocks Genes Linked To Colon Cancer (11/14/07)
Previous research has found that the Wnt signaling pathway is linked to more than 85 percent of sporadic (i. e. not caused by a hereditary defect) colon cancers. Additionally, in vitro studies have shown that resveratrol is capable of blocking the Wnt pathway. The present research showed that in some colon cancer patients who consumed grape powder (which contains resveratrol and possibly other active ingredients), Wnt signaling in biopsied colon tissue was significantly reduced.

Odd protein interaction guides development of olfactory system (10/29/07)
The olfactory system of fruit flies has been shown to develop abnormally when the signaling protein Wnt5 is absent. However, if large amounts of Wnt5 but no Wnt5 receptors called "derailed" are present, development is even more abnormal. Specifically, structures called glomeruli in fruit fly antennal lobes (which are analogous to human olfactory bulbs) grow abnormally when Wnt5 is absent. But if Wnt5 is present in large amounts and there are no derailed receptors, malformed glomeruli develop in locations where they should not be.

Cilia: Small Organelles, Big Decisions (10/3/07)
Research into the development of zebra fish (a favorite of developmental biologists) has shown that organelles called cilia in the cells of developing embryos play a large role in the transduction of Wnt signaling proteins that guide the development process. By blocking the production of three proteins used by cilia, researchers were able to disrupt proper balances in the interpretation of Wnt signals, resulting in developmental defects.

New Insights into the Control of Stem Cells: Keeping the Right Balance (9/15/07)
The Wnt signaling pathway plays a crucial role in embryonic development, cell growth (proliferation), and maturation of cells into specialized cells (differentiation). It is also an important regulator of stem cells. An interaction between Wnt signaling and tyrosine kinases enables the proliferating cells to mature into specialized (differentiated) cells. Normally this interaction strikes a proper balance between proliferation and differentiation. Cancers, such as breast and colon cancer, result when the interaction gets unbalanced. In 90% of human cancers the tumor suppressor APC (adenomatous polypolis coli), one of the core components of the Wnt pathway, is deregulated. This results in excessive amounts of β-catenin, which triggers the onset of breast and colon cancer when it gets into the cell nucleus and affects gene expression.

Reactivating A Critical Gene Lost In Kidney Cancer Reduces Tumor Growth (8/15/07)
Studies of an important tumor-suppressor protein, sFRP-1 (secreted frizzled-related protein 1), in clear cell renal cell carcinoma, the most common type of kidney cancer, may reveal a means to defeat the cancer. sFRP-1 was found to control 13 tumor-promoting genes along the Wnt signaling pathway, which has been linked to a number of cancers, especially colon cancer. Several close relatives of sFRP-1 are also known to affect at least 20 Wnt-related proteins, and up-regulation of members of the sFRP-1 family may be an effective way to control cancers linked to Wnt signaling. In one experiment, increasing sFRP-1 expression in human renal cancer cells was effective, and Wnt regulated oncogenes, such as c-myc, were suppressed compared to untreated cells.

Why Aging Muscles Heal Poorly (8/9/07)
Stem cells normally found in muscle tissue are responsible for repair to muscles damaged by injury or age-related degeneration. But in aged muscle tissue, stem cells tend to produce scar-tissue cells called fibroblasts, instead of normal muscle cells (myoblasts). The overproduction of fibroblasts is a condition known as fibrosis. New research shows that it isn't the age of the muscle stem cells that is the problem, but rather the age of the cellular environment itself, including blood supply to the tissue. The malfunction appears to be a problem with Wnt signaling in the aged environment rather than with the actual stem cells. Muscle stem cells from young mice exhibited the same problems when exposed to an enviroment from older animals.

Related research found that Wnt signaling increased, with detrimental effect, due to age-related deficiency of a hormone called klotho. Klotho seems to inhibit Wnt signaling, and also has some control over insulin sensitivity. However, production of klotho seems to decline with age, possibly leading to age-related problems such as cancer, arterial disease, and insulin resistance.

Not A Relay Race, But A Team Game: New Model For Signal Transduction In Cells (6/27/07)
Details of the inner workings of the Wnt signal transduction process have remained incomplete, but are gradually coming into focus. Member of the Wnt family of proteins may dock with a variety of cell-surface proteins, including LRP6 (low density lipoprotein receptor-related protein 6) and members of the family of G protein-coupled receptors known as Frizzled. After the docking, a signaling cascade is triggered that transmits molecular messages via the cytoplasm to the nucleus. This research shows that the first step after docking involves large protein complexes formed from proteins already known to be part of the signaling pathway, such as phosphorylated LRP6, axin, and Dishevelled (Dvl).

Further reading:

The Wnt Homepage

Regeneration for Repair's Sake

The answer is blowing in the Wnt

Miller on Wnt and Klotho

A hazy shade of Wnt

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Valentine's Day news

Sunday, February 17, 2008

Yeah, yeah, this is late. So?

Every year we read news stories like the following right around this time. Worth thinking about. (By the way, Wagner's Prelude and Liebestod from "Tristan und Isolde" is playing on my computer right now. Synchronicity?)

And here's a little more advice you didn't ask for. First, go see the movie Juno, if you haven't yet. It's good. Really. Second, if you want to understand all this you need to get the perspective of evolutionary psychology. Matt Ridley's The Red Queen: Sex and the Evolution of Human Nature is a good place to start, if you need one.

The Differences in Gender -- Sealed With a Kiss
A kiss, it turns out, is definitely not always just a kiss.

As Valentine's Day approaches, research has begun shedding light on that most basic of all human expressions of love -- the smooch -- which has received surprisingly little scientific scrutiny.

"You'd think there would be a lot of research on kissing behavior. It's so common," said Susan M. Hughes, an assistant professor of psychology at Albright College in Pennsylvania, whose recent study is one of the first to probe snogging in depth. "But there isn't. It's really been ignored."

In fact, much about love and attraction remains mysterious.

"This is a seminal paper," said Helen Fisher, a Rutgers University anthropologist who studies love. "It's remarkable that we don't know more about these things. But love has not really been well studied until recently."

In people, kissing to express affection is almost universal. About 90 percent of human cultures do it.

Carnal Knowledge: They give pleasure; science asks why
There are some natural phenomena whose wonder only deepens upon scientific investigation.

Take the orgasm. Scientists know it involves muscle contractions. They know it makes your pupils expand, and heart rate and blood pressure surge.

But why do orgasms feel good?

I was surprised to find that this is still something of a scientific mystery - though one that a few intrepid researchers are just starting to unravel.

"Really and truly, people don't know," says Julia Heiman, director of the Kinsey Institute for Sex Research and coauthor of Becoming Orgasmic.

Right now, says Heiman, there's a big debate over how the female orgasm evolved. Researchers would also like to know just how different the female kind is from the male.

"Why is it easier for women to have multiple orgasms than it is for men?" Heiman asks. "How does it interact with attachment issues?"

There are lots of other questions, she says, but oddly, in our supposedly sex-obsessed society, it's nearly impossible to get funding for sex research.

Another complication: The orgasm question touches on some profound mysteries about how feelings and consciousness can emerge from the brain.

The Merry Band of Wrigglers


Why Perfect Dates Make Lousy Partners
The best "catches" in dating land may be the worst choices in the long-run, new research shows.

Popular people who monitor themselves carefully in social situations and thereby appear to be the most socially appropriate are often highly sought after as romantic partners, a study finds, but these people show less satisfaction and commitment in relationships than socially-awkward people.

By self-monitoring, people assess how their actions affect others and adjust to fit the appropriateness of the situation. They screen their words and behavior to suit the people around them.

"High self-monitors are social chameleons," said Northwestern University professor of communication studies Michael E. Roloff."And, because they're quick to pick up on social cues, are socially adept and unlikely to say things upsetting to others, they are generally well-liked and sought after."

Self-monitoring is often a helpful attribute.

"Research finds [self-monitors] to be excellent negotiators and far more likely to be promoted at work than their low self-monitoring peers,” Roloff said.

But there’s a downside for high self-monitors when it comes to their romantic relationships.

"High self-monitors may appear to be the kind of people we want to have relationships with, but they themselves are less committed to and less happy in their relationships than low self-monitors," Roloff said.

More: Is Your Dating Partner Happy? With Some People It Is Hard To Know

The next item is about rats, but human folklore suggests it's very likely true for humans as well:

Females love the sweet smell of sexual success
It might be the sweet smell of success or the bitter whiff of despair, but there's something in the odour of a male rat that tells a female whether he's been copulating like crazy or starved of sex for days. And to make matters worse for frustrated males, the females much prefer the smell of road-tested studs.

The next is a fairly interesting article. Unfortunately, the link may turn into a pumpkin rather soon, since Nature isn't very generous with such things. Snarf it quickly if you want it.

Love: You have 4 minutes to choose your perfect mate
Eli Finkel and Paul Eastwick have probably seen more first dates than most. The social scientists at Northwestern University in Evanston, Illinois, have watched hundreds of videos of single people as they participate in a curious, but not unpopular, trend known as speed dating. Two participants spill their souls to each other for a set time, say four minutes, and try to decide whether they might have a future together. When the time is up, they move on to a new partner, sometimes talking to a dozen or more people in a night. ...

From a purely biological standpoint, the success of a partnership hinges mainly on one thing, reproduction. But for humans, who give birth to exceptionally weak, awkward and totally dependent babies, strong pair bonding and the sharing of parental duties can play an important part in the success of their offspring. It is strange, then, that a goal as simple as forming a pair bond could lead to an emotion as complex as romantic love. ...

Since the 1940s, social scientists have brought the tools of their trade to bear on such lofty questions. Finkel and Eastwick are now using some of the newest and most controversial techniques. The fast-paced format of speed dating could be exhilarating, daunting or perhaps even dorky for participants and observers alike. Nevertheless, the researchers say that it could help to reveal some of the mysteries behind that uniquely human emotion — love. Indeed, their research, including a paper published today1, has already started to turn up some surprises.

In the 1940s, when scientists first started to pick at the basis of human attraction, psychologists interviewed single people and asked them what they would value in a partner. Many of the values were the same in both men and women, but two things stood out in survey after survey. Women valued the wealth of their partner much more than men did, and men valued attractiveness more than women did.

These differences can even make sense in evolutionary terms. A woman looking to have children would want the support of a good provider to help her children succeed in life. Men's seemingly superficial preference for beauty was seen as a proxy for health. Symmetry, skin tone and a favourable waist-to-hip ratio could reasonably point to a woman who would not only survive childbirth, but also pass on lots of healthy genes.

More: What Men And Women Say And Do In Choosing Romantic Partners Are Two Different Matters

Other studies on such themes:

Women More Perceptive Than Men In Describing Relationships

'Love Hormone' Promotes Bonding: Could It Treat Anxiety?

Beauty Bias: Can People Love The One They Are Compatible With?

A Sense Of Scarcity: Why It Seems Like All The Good Ones Are Taken

'Hotties' Not So Hot When You're In Love, Online Dating Researchers Find

Probing Women's Response To Male Odor

MicroRNA and cancer

Monday, February 11, 2008

MicroRNAs are 18 to 25 nucleotide, noncoding RNA molecules that have been found to regulate a wide variety of cellular processes. Unusual levels of varions miRNAs have been shown to be diagnostic of pathology in a number of different cancers, such as chronic lymphocytic leukemia, lung cancer and pancreatic cancer.

The first microRNA was discovered in 1993, in C. elegans, but the term microRNA was not introduced until 2001. It took seven years for the second miRNA to be found, also in C. elegans, but many others then followed. Over 500 microRNAs have been identified in the human genome. Over a third of the human genome appears to be regulated by miRNAs orginally found is some mammal or another.

In some cases, changes to gene expression caused by miRNA seem to promote cancer. But in other cases, a miRNA may repress genes that promote cancer or its metastasis.

The p53 anti-cancer gene is affected by some miRNAs, but we will cover that in a separate note. Some miRNAs also seem to be related to cancer stem cells, which is a large topic that merits broader coverage. (See here, here.)

We've already looked at one example (here, and see below) where a gene associated with cancer can affect levels of some miRNAs. Unlike most other findings, this is a case where cancer-related pathology seems to affect miRNA expression levels, rather than the reverse.

One miRNA, in particular, stands out for the variety of genes it may affect. This miRNA is let-7, and it is known to be expressed in the later stages of animal development. Some estimates put the number of human genes affected by let-7 in the hundreds, though most of those may not be related to cancers. Ras is an important cancer-related protein that is suppressed by let-7 (See here, here.) (This report has more on let-7. See also here, here, here, here.)

The miRNA miR-21 has been associated with cancers of the colon, liver, and thyroid.

It's difficult to summarize the following research findings – they involve a variety of cancer types and many different miRNAs. Unusually high or low levels of some miRNAs seem to promote cancer in some cases, but suppress cancer in others. MiRNAs also work in a variety of different ways to affect gene expression and protein activity.

This diversity of effects due to miRNAs may well be the most interesting current finding to come out of research in this area.

Molecules may help predict survival in liver cancer (1/30/08)
In a long-term study of patients with liver cancer, it was found that those with the poorest survival history also had lower levels of 19 specific microRNAs in cancer cells compared to nearby noncancer cells than did patients with significantly better survival. This result is out of a total of 196 different microRNAs whose levels were measured.

Expression Patterns Of MicroRNAs Appear Altered In Colon Cancer, And Associated With Poor Outcomes (1/29/08)
In one cohort of 84 patients with colon cancer 37 different microRNAs were differently expressed in cancer cells compared with noncancer cells. 5 of these miRNAs could reliably discriminate between tumorous and nontumor tissue. The same 5 miRNAs had similar discriminatory powers in a different cohort of 113 patients. For the specific miRNA known as miR-21, high expression levels were associated with poor survival outcomes in both patient cohorts. (High levels of miR-21 are also associated with thyroid cancer, see here, and with liver cancer, see here.)

Two MicroRNAs Promote Spread Of Tumor Cells (1/28/08)
MicroRNAs have been shown in many studies to block translation of tumor suppressor genes. In this study, two specific miRNAs (out of 450 tested) have been shown to transform non-invasive human breast cancer cells into cells that rapidly metastasized in cell cultures and laboratory mice. One of the miRNAs, miR-373, was previously identified as a possible oncogene in testicular cancer. The other miRNA, miR-520c, hasn't previously been associated with cancer, but is similar to miR-373. Both miRNAs are found only in cancer cells. There is evidence that they downregulate the CD44 gene, and that in turn leads to metastasis of non-metastatic tumor cells. In human patients, metastatic cells are found to have higher levels of miR-373 and lower expression of CD44 than non-metastatic tumor cells.

Molecules Might Identify High-risk Acute-leukemia Patients (1/15/08)
In a study of leukemia cells from 122 patients with high- and intermediate-risk acute myeloid leukemia (AML) the same miRNAs could be found in both normal and leukemic cells, but there were differences in levels of various miRNAs present. Two specific miRNAs (miR-191 and miR-199a) were present at abnormally high levels that were clearly associated with patient survival. The same two miRNAs have been previously found to be associated with cancers of the lung, prostate, colon, stomach and breast. Another miRNA (miR-155) was associated with a gene mutation, and high levels of it have been reported in other cancers (see here, here) and to cause leukemia in mice.

Small Molecule Can Prevent Spread Of Breast Cancer, Study Suggests (1/9/08)
Three miRNAs have been found that prevent breast cancer metastasis by interfering with the expression of genes that give cancer cells the ability to proliferate and migrate. Researchers found lower levels of a few miRNAs (miR-335, miR-126 and miR-206) in metastatic cells compared to non-metastatic tumor cells. Testing in mice showed that raising levels of these miRNAs inhibited metastasis. Further analysis showed that miR-126 influences the proliferation rate of metastatic cells, while miR-335 and miR-206 influence the cancer cells' ability to migrate into lungs or bone. miR-335 was found to inhibit expression of SOX4 and TNC genes, which affect cell migration.

More: here, here

Virus Discovered Using Same Tools As Host Cell (12/17/07)
A microRNA expressed in the genome of the Kaposi's Sarcoma Associated Herpesvirus (KSHV) appears to be very similar in stucture and function to a miRNA associated with lymphoma – miR-155, mentioned above. KSHV itself causes a rare skin cancer that disproportionately affects HIV-infected individuals. Both miR-155 and the KSHV miRNA regulate the same genes, including several associated with B cell function and cell cycle regulation. Hence KSHV may promote B cell tumors by repressing one or more of these genes, as miR-155 seems to do.

Scientists Identify And Repress Breast Cancer Stem Cells In Mouse Tissue (12/17/07)
Since 2001 stem-like cells that appear to initiate cancer development have been discovered in breast, lung, brain and colon tissues, as well as in the blood. A microRNA (let-7) has now been found that can help to identify such cancer stem cells in breast cancer tissue of mice. Further tests indicate that let-7 can attack and eliminate these cancer progenitors.

MicroRNA Regulates Cancer Stem Cells: Could Lead To Treating Cancer As A Whole (12/13/07)
Another study involving the interaction of miRNA let-7 and cancer stem cells was conducted independently and published just before the one described above. In this study, researchers found a way to grow large quantities of tumor stem cells by growing human breast cancer cells in immunosuppressed mice. They found that these stem cells contained low amounts of several miRNAs, compared to more mature tumor cells or stem cells that had differentiated in culture. When let-7 was activated in these cells, they lost their ability to self-renew and began to differentiate. They also became less able to form tumors in mice or to metastasize. It appears that let-7 did this by switching off two cancer-related genes: the oncogene Ras, and HMG2A (see here, here).

Silencing Small But Mighty Cancer Inhibitors (12/10/07)
As discussed here, the important transcription factor Myc, which is overexpressed in many cancers, can also stop the production of at least 13 microRNAs. Some of these miRNAs have an inhibiting effect on cancer. In some cases re-introducing repressed miRNAs into Myc-containing cancer cells suppressed tumor growth in mice. So repression of miRNAs may be another pathway through which overexpression of Myc promotes cancer. The research involved lymphoma cells in mice, and showed that Myc repressed the miRNAs by directly attaching to the DNA at the miRNA genes.

Scientists discover new role for miRNA in leukemia (12/10/07)
A microRNA has been found to play a new role in the development of cancer, in this case chronic myeloid leukemia (CML). The miRNA is miR-328, and normally it is able to directly bind to a certain protein, inhibiting the activity of that protein. But if levels of miR-328 become abnormally low, the protein prevents white blood cells maturing as they should. The result is the build-up of immature white blood cells and entrance to what is called the "blast-crisis" phase of the disease.

Cellular Pathway Identified That Makes Prostate Cancer Fatal (11/27/07)
Expression levels of microRNAs were measured in samples of prostate cancer cells. Five different miRNAs were found to have unusual expression levels. One of these, miR-125b, was found at high levels in both androgen-dependent and the more dangerous androgen-independent prostate cancer cells.

Earlier research reports:

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Human gene count drops again

Monday, February 4, 2008

Before the human genome was sequenced and the results published in February 2001, some biologists speculated that there might be 100,000 or more different genes. Later in 2001 the estimated numbers were still sometimes between 60,000 and 90,000. (See here, here, here.) More conservative estimates at the time were around 35,000, and that gradually fell to about 25,000 over the next several years.

The problem is that there are no dependably unambiguous markers within the DNA itself to identify where a potentially active gene starts and ends. Remember that every strand of DNA has a sense of directionality established by the two ends of the strand, which are chemically distinct and called the 5′ and 3′ ends. The enzymes that transcribe DNA into RNA always read from the 5′ to the 3′ end, so the start of a gene is closer to the former than the latter. Every gene also has a "promoter" region, which is a short DNA sequence located in the 5′ direction ("upstream") from the gene itself. Transcription factors attach themselves to the promoter region in order to enable gene transcription.

Although it is relatively easy to recognize promoter regions, they can be located far from the gene itself, and it's still often difficult to predict where the corresponding gene (if any) actually begins. See here for much more detail on gene finding.

Initial high estimates of gene numbers were made indirectly, based on the number of distinct proteins that existed in human cells. It was known that this number was at least 100,000 or more, possibly a lot more. But this was misleading, because it wasn't well understood that a single gene could code for multiple proteins, through the process of alternative splicing.

One clue to the actual start of a gene is the presence of a start codon. This is a specific three-letter sequence (ATG). In eukaryotes this codon encodes the amino acid methionine, but it is usually preceded by a "5′ untranslated region" ("5′ UTR") as further identification. The end of a gene is marked by a stop codon. The portion of DNA between the start and stop codons is called an "open reading frame" (ORF). Another condition needed is a sequence of enough codons (100, i. e. 300 nucleotides) in order to encode a working protein.

It is still not true that all sufficiently long open reading frames correspond to actual genes. There are various heuristics used to identify ORFs that do not really correspond to genes, but potential uncertainty remains, because there's no unambiguous way to tell from the DNA itself that a particular ORF actually corresponds to a working gene. It might instead, for example, have been an actual gene in some distant human ancestor but is no longer functional in humans. (Such things are known as "pseudogenes".)

But now that we know the DNA sequences of various other mammals, it is possible to identify more pseudogenes in human DNA, further reducing the total count of actual functioning genes.

Human Gene Count Tumbles Again
Estimates of the number of genes in the human genome have ranged wildly over the past two decades, from 20,000 all the way up to 150,000. By the time the working draft of the human genome was published in 2001, the best approximation stood at 35,000, yet even that number has fallen. A new analysis, one that harnesses the power of comparing genome sequences of various organisms, now reveals that the true number of human genes is about 20,500, thousands fewer than what is currently listed in human gene catalogs.

The initial clue that not all sequences among the 25,000 that had been settled upon as "real" human genes actually were such is that many did not correspond to genes identified in the mouse genome. This was suspicious, since working, useful genes in the common ancestor of humans and mice ought to be conserved in both later species.

A supposed human gene which did not correspond to a mouse gene might have appeared in the time since humans and mice diverged from their common ancestor, or the gene might have been lost by mice (but not humans) sometime after the common ancestor. On the other hand it might not be a real human gene at all (having lost functionality along the way). One method to distinguish between these two cases is to check whether an analogue of the supposed gene could be found in a primate genome, since the primates whose genomes have been cataloged (macaques and chimpanzees) are much more closely related to humans than mice are.

Ultimately, almost 5000 pseudogenes have been removed from the earlier list of 25,000 human genes. Sequences of human DNA that appear to be genes but do not correspond to genes in mice and dogs, yet do correspond to genes in macaques and chimpanzees, are considered real. The remainder, mostly, are considered pseudogenes:
To distinguish such misidentified genes from true ones, the research team, led by Clamp and Broad Institute director Eric Lander, developed a method that takes advantage of another hallmark of protein-coding genes: conservation by evolution. The researchers considered genes to be valid if and only if similar sequences could be found in other mammals – namely, mouse and dog. Applying this technique to nearly 22,000 genes in the Ensembl gene catalog, the analysis revealed 1,177 “orphan” DNA sequences. These orphans looked like proteins because of their open reading frames, but were not found in either the mouse or dog genomes.

Although this was strong evidence that the sequences were not true protein-coding genes, it was not quite convincing enough to justify their removal from the human gene catalogs. Two other scenarios could, in fact, explain their absence from other mammalian genomes. For instance, the genes could be unique among primates, new inventions that appeared after the divergence of mouse and dog ancestors from primate ancestors. Alternatively, the genes could have been more ancient creations — present in a common mammalian ancestor — that were lost in mouse and dog lineages yet retained in humans. ...

After extending the analysis to two more gene catalogs and accounting for other misclassified genes, the team’s work invalidated a total of nearly 5,000 DNA sequences that had been incorrectly added to the lists of protein-coding genes, reducing the current estimate to roughly 20,500.

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Something to think about on Superbowl Sunday

Friday, February 1, 2008

This seems just about right:

Sports Machismo May Be Cue To Male Teen Violence


"Sports such as football, basketball, and baseball provide players with a certain status in society," said Derek Kreager, assistant professor of sociology in the Crime, Law, and Justice program. "But football and wrestling are associated with violent behavior because both sports involve some physical domination of the opponent, which is rewarded by the fans, coaches and other players."

Probably has something to do with military recruiting too:
The researcher found that, compared with non-athletes, football players and wrestlers face higher risks of getting into a serious fight by over 40 per cent. High-contact sports that are associated with aggression and masculinity increase the risk of violence, he concluded.

And if you want a little music to go along with this information, try this.

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