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Biological Mathematics - Biomatics.org

Biological Mathematics

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Principia BioMathematica

The notion that DNA and proteins form some sort of computational network goes back to at least as far as the 1970’s.  
In 1982, Richard Feynman referred to the idea of a 'quantum computer', a computer that uses the effects of quantum mechanics.  In 1994, Len Adelman demonstrated his- computation in a test tube- based on DNA splicing mechanisms.  Currently, many uses of the term “biological mathematics” refer to Adelman’s splicing techniques.  More recently "Intramolecular computation" has been added to this list, i.e. computation occurring within single molecules e.g. The Amino Acid Code and The Histone Code, as well as any other techniques that biological systems may use that qualify as mathematical in nature.

Computation implies some form of mathematics.  Adelman’s experiment solved the so-called traveling salesman or shortest path problem, at least for a limited set of data.  DNA and protein networks respond to complex logical environments involving decisions based on the absence or presence of different conditions, molecules or organisms in the cellular environment.  At the level of the brain, extremely complex mathematical processing must be occurring.  Artificial Intelligence techniques give many potential models- Decision theory, Statistical pattern recognition, Image processing techniques- to name just a few.

The tetrahedral geometry of the carbon atom is at the center or kernel of biological mathematics(smart molecules).  Carbon atoms readily form chains amongst themselves (as well as other classes of atoms) by the sharing of electrons. The interactions of these neighboring covalent bonds result in switching elements similar in potential function to their man made digital counterparts.  The result is that as few as two or three atoms acting in concert have a wealth of mathematical and logical processing capabilities. (Computational Structures in Non-Coding DNA and the Histone Code)






The Theory of Predication in Aristotle's Categories





From: Michael Frede - Categories in Aristotle. In Studies in Aristotle. Edited by O'Meara Dominic. Washington: Catholic University Press 1981. pp. 1-25
Reprinted in: M. Frede - Essays in Ancient Philosophy - Minneapolis, University of Minnesota Press, pp. 29-48.




In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. Although this appears to be a simple idea, sets are one of the most fundamental concepts in modern mathematics. The study of the structure of possible sets, set theory, is rich and ongoing. Having only been invented at the end of the 19th century, set theory is now a ubiquitous part of mathematics education, being introduced from primary school in many countries. Set theory can be viewed as the foundation upon which nearly all of mathematics can be derived.











In mathematics, the concept of a relation or relationship is a generalization of 2-place relations, such as the relation of equality, denoted by the sign "=" in a statement like "5 + 7 = 12," or the relation of order, denoted by the sign "<" in a statement like "5 < 12". Relations that involve two places or roles are called binary relations by some and dyadic relations by others, the latter being historically prior but also useful when necessary to avoid confusion with binary (base 2) numerals.


The next step up is to consider relations that can involve more than two places or roles, but still a finite number of them. These are called finite place or finitary relations. A finitary relation that involves k places is variously called a k-ary, a k-adic, or a k-dimensional relation. The number k is then called the arity, the adicity, or the dimension of the relation, respectively.

Numerical Analysis






Animations for Numerical Methods and Numerical Analysis


Monte Carlo method

The Monte Carlo method can be illustrated as a game of battleship. First a player makes some random shots. Next the player applies algorithms (i.e. a battleship is four dots in the vertical or horizontal direction). Finally based on the outcome of the random sampling and the algorithm the player can determine the likely locations of the other player's ships.


Monte Carlo simulation methods are especially useful in studying systems with a large number of coupled degrees of freedom, such as fluids, disordered materials, strongly coupled solids, and cellular structures (see cellular Potts model). More broadly, Monte Carlo methods are useful for modeling phenomena with significant uncertainty in inputs, such as the calculation of risk in business. These methods are also widely used in mathematics: a classic use is for the evaluation of definite integrals, particularly multidimensional integrals with complicated boundary conditions.


Orthonormal Basis

A subset {v_1,...,v_k} of a vector space V, with the inner product <,>, is called orthonormal if <v_i,v_j>=0 when i!=j. That is, the vectors are mutually perpendicular. Moreover, they are all required to have length one: <v_i,v_i>=1.

An orthonormal set must be linearly independent, and so it is a vector space basis for the space it spans. Such a basis is called an orthonormal basis.

The simplest example of an orthonormal basis is the standard basis e_i for Euclidean space R^n. The vector e_i is the vector with all 0s except for a 1 in the ith coordinate. For example, e_1=(1,0,...,0). A rotation (or flip) through the origin will send an orthonormal set to another orthonormal set. In fact, given any orthonormal basis, there is a rotation, or rotation combined with a flip, which will send the orthonormal basis to the standard basis. These are precisely the transformations which preserve the inner product, and are called orthogonal transformations.

Usually when one needs a basis to do calculations, it is convenient to use an orthonormal basis. For example, the formula for a vector space projection is much simpler with an orthonormal basis. The savings in effort make it worthwhile to find an orthonormal basis before doing such a calculation. Gram-Schmidt orthonormalization is a popular way to find an orthonormal basis.

Another instance when orthonormal bases arise is as a set of eigenvectors for a symmetric matrix. For a general matrix, the set of eigenvectors may not be orthonormal, or even be a basis.

A wave function or wavefunction is a mathematical tool used in quantum mechanics to describe any physical system. It is a function from a space that maps the possible states of the system into the complex numbers. The laws of quantum mechanics (i.e. the Schrödinger equation) describe how the wave function evolves over time. The values of the wave function are probability amplitudes — complex numbers — the squares of the absolute values of which give the probability distribution that the system will be in any of the possible states.

The electron probability density for the first few hydrogen atom electron orbitals  form an orthonormal basis for the wave function of the electron.

Unsupervised Learning


Unsupervised learning is a method of machine learning where a model is fit to observations. It is distinguished from supervised learning by the fact that there is no a priori output. In unsupervised learning, a data set of input objects is gathered. Unsupervised learning then typically treats input objects as a set of random variables. A joint density model is then built for the data set.

Unsupervised learning can be used in conjunction with Bayesian inference to produce conditional probabilities (i.e. supervised learning) for any of the random variables given the others. A holy grail of unsupervised learning is the creation of a factorial code of the data, i. e., a code with statistically independent components. Later supervised learning usually works much better when the raw input data is first translated into a factorial code.

Unsupervised learning is also useful for data compression: fundamentally, all data compression algorithms either explicitly or implicitly rely on a probability distribution over a set of inputs.

Another form of unsupervised learning is clustering, which is sometimes not probabilistic. Also see formal concept analysis.


Distance measure

An important step in any clustering is to select a distance measure, which will determine how the similarity of two elements is calculated. This will influence the shape of the clusters, as some elements may be close to one another according to one distance and further away according to another. For example, in a 2-dimensional space, the distance between the point (x=1, y=0) and the origin (x=0, y=0) is always 1 according to the usual norms, but the distance between the point (x=1, y=1) and the origin can be 2,\sqrt 2 or 1 if you take respectively the 1-norm, 2-norm or infinity-norm distance.

Common distance functions:

  • The Manhattan distance (also called taxicab norm or 1-norm)
  • The maximum norm
  • The Mahalanobis distance corrects data for different scales and correlations in the variables
  • The angle between two vectors can be used as a distance measure when clustering high dimensional data. See Inner product space.
  • The Hamming distance (sometimes edit distance) measures the minimum number of substitutions required to change one member into another.
  • Some notions of Semantic relatedness are distance functions. These include distances based on databases such as wordnet and search engines, and distances learned from machine-learned semantic analysis of a corpus.

Cardinal and Ordinal numbers

One view is that the core of mathematics is based upon two simple questions based on practical needs.

. How many?

. How much?

This is the cardinal number viewpoint. 

Another view is that mathematics may have an even earlier basis based on ordinals used to establish pecking orders and rank. Such basic questions are:

. Who eats first, second, etc?

. What comes first, etc?

This is the ordinal number viewpoint. 

One to One Correspondence

The notion of one-to-one correspondence is  fundamental to counting. When we count out a set of cards, we say, 1, 2, 3, ... , 52, and as we say each number we lay down a card. Each number corresponds to a card. Technically, we can say that we have put the cards in the deck and the numbers from 1 to 52 in a one-to-one correspondence with each other.

In abstract algebra, a homomorphism is a (one to one) structure-preserving map between two algebraic structures (such as groups, rings, or vector spaces). The word homomorphism comes from the Greek language: homos meaning "same" and morphe meaning "shape". Note the similar root word "homoios," meaning "similar," which is found in another mathematical concept, namely homeomorphisms.

In abstract algebra, an isomorphism (Greek: ison "equal", and morphe "shape") is a (one to one and onto) bijective map f such that both f and its inverse f −1 are homomorphisms, i.e., structure-preserving mappings.


Algebra is a branch of mathematics concerning the study of structure, relation and quantity. The name is derived from the treatise written by the Persian[1] mathematician, astronomer, astrologer and geographer, Muhammad bin Mūsā al-Khwārizmī titled (in Arabic الكتاب الجبر والمقابلة ) Al-Kitab al-Jabr wa-l-Muqabala (meaning "The Compendious Book on Calculation by Completion and Balancing"), which provided symbolic operations for the systematic solution of linear and quadratic equations







Sets: Rather than just considering the different types of numbers, abstract algebra deals with the more general concept of sets: a collection of all objects (called elements) selected by property, specific for the set. All collections of the familiar types of numbers are sets. Other examples of sets include the set of all two-by-two matrices, the set of all second-degree polynomials (ax² + bx + c), the set of all two dimensional vectors in the plane, and the various finite groups such as the cyclic groups which are the group of integers modulo n. Set theory is a branch of logic and not technically a branch of algebra.


Binary operations: The notion of addition (+) is abstracted to give a binary operation, * say. The notion of binary operation is meaningless without the set on which the operation is defined. For two elements a and b in a set S a*b gives another element in the set; this condition is called closure). Addition (+), subtraction (-), multiplication (×), and division (÷) can be binary operations when defined on different sets, as is addition and multiplication of matrices, vectors, and polynomials.


Identity elements: The numbers zero and one are abstracted to give the notion of an identity element for an operation. Zero is the identity element for addition and one is the identity element for multiplication. For a general binary operator * the identity element e must satisfy a * e = a and e * a = a. This holds for addition as a + 0 = a and 0 + a = a and multiplication a × 1 = a and 1 × a = a. However, if we take the positive natural numbers and addition, there is no identity element.


Inverse elements: The negative numbers give rise to the concept of inverse elements. For addition, the inverse of a is -a, and for multiplication the inverse is 1/a. A general inverse element a-1 must satisfy the property that a * a-1 = e and a-1 * a = e.


Associativity: Addition of integers has a property called associativity. That is, the grouping of the numbers to be added does not affect the sum. For example: (2+3)+4=2+(3+4). In general, this becomes (a * b) * c = a * (b * c). This property is shared by most binary operations, but not subtraction or division or octonion multiplication .


Commutativity: Addition of integers also has a property called commutativity. That is, the order of the numbers to be added does not affect the sum. For example: 2+3=3+2. In general, this becomes a * b = b * a. Only some binary operations have this property. It holds for the integers with addition and multiplication, but it does not hold for matrix multiplication or quaternion multiplication .

Molecular Algebra



Twenty Elementary Algebras

The above image is a Moncznik (Perry Moncznik) multiplication table (in the Abstract Algebra sense) where the elements being "multiplied" are the different possible conformational states of a single amino acid. The twenty amino acids can each be represented by such a table and thus form twenty elemental algebras, which it could be argued, may in some sense form the basis from which a large portion of mathematics arises.


Variables, Expressions, and Equations


A variable is a symbol that represents a number. Usually we use letters such as n, t, or x for variables. For example, we might say that s stands for the side-length of a square. We now treat s as if it were a number we could use. The perimeter of the square is given by 4 × s. The area of the square is given by s × s. When working with variables, it can be helpful to use a letter that will remind you of what the variable stands for: let n be the number of people in a movie theater; let t be the time it takes to travel somewhere; let d be the distance from my house to the park. 



An expression is a mathematical statement that may use numbers, variables, or both.


The following are examples of expressions:



3 + 7

2 × y + 5

2 + 6 × (4 - 2)

z + 3 × (8 - z)



An equation is a statement that two numbers or expressions are equal. Equations are useful for relating variables and numbers. Many word problems can easily be written down as equations with a little practice. Many simple rules exist for simplifying equations




A bijective function.

In mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property that, for every y in Y, there is exactly one x in X such that
f(x) = y.

Alternatively, f is bijective if it is a one-to-one correspondence between those sets; i.e., both one-to-one (injective) and onto (surjective).[1] (See also Bijection, injection and surjection.)

For example, consider the function succ, defined from the set of integers to , that to each integer x associates the integer succ(x) = x + 1. For another example, consider the function sumdif that to each pair (x,y) of real numbers associates the pair sumdif(x,y) = (x + y, x − y).

A bijective function is also called a permutation. This is more commonly used when X = Y. It should be noted that one-to-one function means one-to-one correspondence (i.e., bijection) to some authors, but injection to others. The set of all bijections from X to Y is denoted as XY.

Bijective functions play a fundamental role in many areas of mathematics, for instance in the definition of isomorphism (and related concepts such as homeomorphism and diffeomorphism), permutation group, projective map, and many others.

Composition and Inverses

A function f is bijective if and only if its inverse relation f −1 is a function. In that case, f −1 is also a bijection.

The composition g o f of two bijections f XY and g YZ is a bijection. The inverse of g o f is (g o f)−1 = (f −1o (g−1).



A bijection composed of an injection and a surjection.

On the other hand, if the composition g o f of two functions is bijective, we can only say that f is injective and g is surjective.

A relation f from X to Y is a bijective function if and only if there exists another relation g from Y to X such that g o f is the identity function on X, and f o g is the identity function on Y. Consequently, the sets have the same cardinality.


Mathematical structure


In mathematics , a structure on a set, or more generally a type, consists of additional mathematical objects that in some manner attach to the set, making it easier to visualize or work with, or endowing the collection with meaning or significance.

A partial list of possible structures are measures, algebraic structures (groups, fields, etc.), topologies, metric structures (geometries), orders, equivalence relations, and differential structures.

Sometimes, a set is endowed with more than one structure simultaneously; this enables mathematicians to study it more richly. For example, an order induces a topology. As another example, if a set both has a topology and is a group, and the two structures are related in a certain way, the set becomes a topological group.

Mappings between sets which preserve structures (so that structures in the domain are mapped to equivalent structures in the codomain) are of special interest in many fields of mathematics. Examples are homomorphisms, which preserve algebraic structures; homeomorphisms, which preserve topological structures; and diffeomorphisms, which preserve differential structures.


Discrete Mathematics



For contrast, see continuum, topology, and mathematical analysis.

Discrete mathematics includes the following topics:

Molecular Geometry

The term geometric primitive in computer graphics and CAD systems is used in various senses, with common meaning of atomic geometric objects the system can handle (draw, store). Sometimes the subroutines that draw the corresponding objects are called "geometric primitives" as well. The most "primitive" primitives are point and straight line segment, which were all that early vector graphics systems had.




Commonly used geometric primitives include:



The specific three dimensional arrangement of atoms in molecules is referred to as molecular geometry. Molecular geometry is associated with the specific orientation of atoms as a result of bonding and non bonding electrons about the central atom. A careful analysis of electron pairs will usually result in correct molecular geometry determinations. In addition, the simple writing of Lewis diagrams which show the electron arrangements can also provide important clues for the determination of molecular geometry. Molecules with no lone electron pairs: Molecular geometry has its basis in the electron pair geometry of a molecule. If the molecule has all electron pairs bonded to atoms, then the the molecular geometry is identical with the electron pair geometry. This is a common occurrence


An example of trigonal bipyramid molecular geometry that results from five electron pair geometry is PCl5. The phosphorus has 5 valence electrons and thus needs 3 more electrons to complete its octet. However this is an example where five chlorine atoms present and the octet is expanded.

The Lewis diagram is as follows:
Cl = 7 e- x 5 = 35 e-
P = 5 e- = 5 e-

Total = 40 e-

The Chlorine atoms are as far apart as possible at nearly 90o and 120obond angle. This is trigonal bipyramid geometry.

Trigonal bipyramid geometry is characterized by 5 electron pairs. 


An example of octahedral molecular geometry that results from six electron pair geometry is SF6. The sulfur atom has 6 valence electrons. However this is an example where six fluoride atoms are present and the octet is expanded.

The Lewis diagram is as follows:
F = 7 e- x 6 = 42 e-
S = 6 e- = 6 e-

Total = 48 e-

The fluorine atoms are as far apart as possible at nearly 90o bond angle in all directions. This is octahedral geometry.

Octahedral geometry is characterized by 6 electron pairs.

Trigonal planar 
Trigonal pyramid. 





Constructive solid geometry (CSG) is a technique used in solid modeling. CSG is often, but not always, a procedural modeling technique used in 3D computer graphics and CAD. Constructive solid geometry allows a modeler to create a complex surface or object by using Boolean operators to combine objects. Often CSG presents a model or surface that appears visually complex, but is actually little more than cleverly combined or decombined objects. (In some cases, constructive solid geometry is performed on polygonal meshes, and may or may not be procedural and/or parametric.)




Operations in constructive solid geometry
Boolean union Boolean difference Boolean intersection
Demonstration of CSG Union Demonstration of CSG Difference Demonstration of CSG Intersection
The merger of two objects into one. The subtraction of one object from another. The portion common to both objects.


The Centriole

The Mother Centriole Plays an Instructive Role in Defining Cell Geometry 

"The centriole is unique among cellular structures in its complexity, chirality, stability, and templated replication, and these features make it an ideal hub around which to organize and propagate particular aspects of cellular geometry."

The Telomere Counting Mechanism




The Telomere Code

Experimental design and data analysis


 Control theory




The Microtubule Code

anti-tubulin and DAPI

In interphase, the DNA is neatly contained within the nucleus and is not condensed into chromosomes. Microtubules are radially arrayed from the center of the cell and if this cell was not fixed and dead the microtubules would be highly dynamic, shrinking and growing from their ends. When a cell is ready to divide it will replicate both its DNA and cellular contents and then split into two in the process known as mitosis. The images here come from Xenopus XL-177 cells.

Tubulin is shown in green, DNA in blue.

Dynein and kinesin motor proteins transport cellular cargoes toward opposite ends of microtubule tracks. In neurons, microtubules are abundantly decorated with microtubule-associated proteins (MAPs) such as tau. Motor proteins thus encounter MAPs frequently along their path. Dynein tends to reverse direction, whereas kinesin tends to detach at patches of bound tau. The differential modulation of dynein and kinesin motility suggests that MAPs can spatially regulate the balance of microtubule-dependent axonal transport. Does a "microtubule code" regulate activity of MAPs? 


Micro-fibo2.jpg http://www.quantumconsciousness.org/views/Topological.html

Cylindrical cellular automata

Source: Comm. Math. Phys. Volume 118, Number 4 (1988), 569-590. PDF File (2172 KB) 

This paper is concerned with the analysis of one-dimensional cellular automata with periodic boundary conditions. Such an automaton may be viewed as a lattice of sites on a cylinder of specified size n evolving according to a local interaction rule…


Hexagonal Numbers


A polygonal number and 6-polygonal number of the form n(2n-1). The first few are 1, 6, 15, 28, 45, ... (Sloane's A000384). The generating function for the hexagonal numbers is given by


Every hexagonal number is a triangular number since


In 1830, Legendre (1979) proved that every number larger than 1791 is a sum of four hexagonal numbers, and Duke and Schulze-Pillot (1990) improved this to three hexagonal numbers for every sufficiently large integer.

There are exactly 13 positive integers that cannot be represented using four hexagonal numbers, namely 5, 10, 11, 20, 25, 26, 38, 39, 54, 65, 70, 114, and 130 (Sloane's A007527; Guy 1994a).

Similarly, there are only two positive integers that cannot be represented using five hexagonal numbers, namely:

11 = 1+1+1+1+1+6
26 = 1+1+6+6+6+6.

Every positive integer can be represented using six hexagonal numbers.

SEE ALSO: Figurate Number, Hex Number, Heptagonal Hexagonal Number, Hexagonal Pentagonal Number, Octagonal Hexagonal Number, Triangular Number


Duke, W. and Schulze-Pillot, R. "Representations of Integers by Positive Ternary Quadratic Forms and Equidistribution of Lattice Points on Ellipsoids." Invent. Math. 99, 49-57, 1990.

Guy, R. K. "Every Number Is Expressible as the Sum of How Many Polygonal Numbers?." Amer. Math. Monthly 101, 169-172, 1994a.

Guy, R. K. "Sums of Squares." §C20 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 136-138, 1994b.

Legendre, A.-M. Théorie des nombres, 4th ed., 2 vols. Paris: A. Blanchard, 1979.

Sloane, N. J. A. Sequences A000384/M4108 and A007527/M3739 in "The On-Line Encyclopedia of Integer Sequences."

Weisstein, Eric W. "Hexagonal Number." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/HexagonalNumber.html

Cellular Automata and the Game of Life in the Hexagonal Grid

Only one true hexagonal Game of Life has been found. The rule 3/2 supports a glider and also stabilizes. (The rule 3,5/2 and 3,5,6/2 also behave silimarly.) (3/2,4,5) barely doesn't qualify, as random patterns never stabilize. 


Polyglutamylation and polyglycylation are two posttranslational polymodifications that have initially been discovered on tubulin

Highly desirable features in a network of parallel machines  
  •    minimal communication cost 

  •    efficient routing

  •    capability of embedding topological Data structures such as

  •    Ring

  •   Linear array

  •   Tree

  •   Mesh




The Krebs Cycle

The Kreb's cycle converts pyruvate to CO2 and reducing energy (NADH and FADH2) and phosphorylated energy (GTP).

2 pyruvate + 2 GDP + 2 H3PO4 + 4 H2O + 2 FAD + 8 NAD+ ----> 6 CO2 + 2 GTP + 2 FADH2 + 8 NADH

Stoichiometry Matrix-


Cellular Automata, Transition Algebras and The Genetic State Vector

The Cellular machinery can be seen as the interaction of these 3 distinct mathematical objects acting in a loop. 

The concept of "hypercellular automata", or multilayered cellular automata, has recently been proposed by [Bandini, 1995] and [Bandini et al., 1996], as a particular case of multilayered automata network. A hierarchical structure is defined through a hypergraph, i.e. a graph composed by vertices and arcs, where each vertex is in turn a hypergraph. The multilayered automata network is directly obtained from this structure by introducing status attributes and transition functions.  Two-level multilayered cellular automata have been developed and employed to model biological systems: the first level constitutes a bidimensional cellular space (diffusion space), while at the second level a totally connected graph corresponds to each cell (first-level vertex) to generate an intrinsically parallel and local reaction space.
  Figure 1.  The hypercellular automaton

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