A

Active transport  is the mediated transport of biochemicals, and other atomic/molecular substances, across membranes. Unlike passive transport, this process requires chemical energy. In this form of transport, molecules move against either an electrical or concentration gradient (collectively termed an electrochemical gradient). This is achieved by either altering the affinity of the binding site or altering the rate at which the protein changes conformations.

B

Brownian motion In 1827 the English botanist Robert Brown noticed that pollen grains suspended in water jiggled about under the lens of the microscope, following a zigzag path. Even more remarkable was the fact that pollen grains that had been stored for a century moved in the same way. Nowadays, the term Brownian motion refers to either the physical phenomenon that minute particles, immersed in a fluid, move about randomly; or the mathematical models used to describe those random movements. The mathematical model can also be used to describe many phenomena not resembling (other than mathematically) the random movements of minute particles.

C

Cellular Potts Model  In statistical mechanics, the Potts model, a generalization of the Ising model, is a model of interacting spins on a crystalline lattice. By studying the Potts model, one may gain insight into the behaviour of ferromagnets and certain other phenomena of solid state physics. The strength of the Potts model is not so much that it models these physical systems well; it is rather that the one-dimensional case is exactly solvable, and that it has a rich mathematical formulation that has been studied extensively.

The model is named after Renfrey B. Potts who described the model near the end of his 1952 Ph.D. thesis. The model was related to the "planar Potts" or "clock model", which was suggested to him by his advisor Cyril Domb. The Potts model is sometimes known as the Ashkin-Teller model (after Julius Ashkin and Edward Teller), as they considered a four component version in 1943.

Chemical potential of a thermodynamic system is the amount by which the energy of the system would change if an additional particle were introduced, with the entropy and volume held fixed. If a system contains more than one species of particle, there is a separate chemical potential associated with each species, defined as the change in energy when the number of particles of that species is increased by one.

Chemotaxis is directional movement (migration) of biological cells or organisms in response to concentration gradients of chemicals, whereby the cells are attracted or repelled by substances exhibiting chemical properties.

Conditional probability is the probability that an event will occur, given that another event has already occurred. The probability that Event A will occur on the condition that Event B has already occurred, usually denoted P(A|B).

Current is the rate of flow of an electric charge. Current is measured in amps.

D

Deterministic means referring to events that have no random or probabilistic aspects but proceed in a fixed predictable fashion.

Dynamic equilibrium The state of system wherein the output continually changes but remains within fairly narrow bounds. The output is characterized by a mean value and a bounded range around that mean value. Example: water liquid in balance with the vapor, where the number of molecules leaving the liquid at any moment is approximately (but probably not exactly) equal to the number leaving the vapor for the liquid. But over any interval of time, these two quantities are equal. We say that condensation and evaporation are in dynamic equilibrium at saturation.

Dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. The mathematical models used to describe the swinging of a clock pendulum, the flow of water in a pipe, or the number of fish each spring in a lake are examples of dynamical systems.

E

A number is called an eigenvalue of a matrix if there exists a nonzero vector such that the matrix times the vector is equal to the same vector multiplied by the eigenvalue. Such equation is called eigen-equation. This vector is then called the eigenvector associated with the eigenvalue.

Energy  is the capacity of a physical system to do work; the units of energy are joules or ergs.

Entropy is a measure of the level of disorder or randomness in a closed system. It can be thought of either in the sense of thermodynamic/metabolic processes or the increasing molecular disorder in a structure. It can be thought of as the same process by which erosion occurs when soil is exposed to rain and wind.

Equilibrium is a stable situation in which forces cancel one another

F

Ferromagnetism is a phenomenon by which a material can exhibit a spontaneous magnetization, and is one of the strongest forms of magnetism. It is responsible for most of the magnetic behavior encountered in everyday life and, along with ferrimagnetism, is the basis for all permanent magnets (as well as the metals that are noticeably attracted to them).

Force is that which can impose a change of velocity on a material body. This concept first appeared in Newton's second law of motion of classical mechanics, and is usually expressed by the equation: Force has been, however, more accurately defined as being the derivative of momentum.

Function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the codomain, not to be confused with the range). The concept of a function is fundamental to virtually every branch of mathematics and every quantitative science.

H

Haptotaxis is the directed movement of cell motility or outgrowth, e.g. in the case of axonal outgrowth, either up or down a gradient of cellular adhesion sites. These gradients are often present in biomaterials seeded with cells of interest, where the gradients are established by altering the concentration of adhesion sites in a polymer.

Hypothesis test is a test based on statistical data, where you must chose between two alternatives.

I

Interphase is the period in the cell cycle when DNA is replicated in the nucleus; followed by mitosis. To find more about interphase, click the following link.

Iteration is the repetition of a process, typically within a computer program. It can be used both as a general term, synonymous with repetition, and to describe a specific form of repetition with a mutable state.

Initial state is the first element of a sequence of states which constitute a computation.

The Ising model, named after the physicist Ernst Ising, is a mathematical model in statistical mechanics. It can be represented on a graph where its configuration space is the set of all possible assignments of +1 or −1 to each vertex of the graph. The graph can exhibit periodic boundary conditions or free space boundary conditions depending on the system being modelled. To complete the model, a function, E(e) must be defined, giving the difference between the energy of the "bond" associated with the edge when the spins on both ends of the bond are opposite and the energy when they are aligned. It is also possible to include an external magnetic field. The Ising model is also used as a model of a simple liquid. Most physicists agree that the Ising model can be used interchangeably with atomistic models of simple liquids. In fact due to the computational efficiency the Ising model stands to render atomistic solvents obsolete by early 2007.

L

The lattice points are points in the coordinate plane with real coordinates.

M

Magnetic field is an entity produced by moving electric charges (electric currents) that exerts a force on other moving charges. (The quantum-mechanical spin of a particle produces magnetic fields and is acted on by them as though it were a current; this accounts for the fields produced by "permanent" ferromagnets.)

Magnetism is a phenomenon by which materials exert an attractive or repulsive force on other materials. Some well known materials that exhibit easily detectable magnetic properties are iron, some steels, and the mineral lodestone; however, all materials are influenced to one degree or another by the presence of a magnetic field, although in most cases the influence is too small to detect without special equipment.

Mathematical process is a calculation by mathematical methods; "the problems at the end of the chapter demonstrated the mathematical processes involved in the derivation"; "they were learning the basic operations of arithmetic".

Matrix is a two-dimensional rectangle with row and column values.

Matrix multiplication is a way to multiply matrices. Click here to find out how to do it.

A Markov chain is a discrete stochastic process in which the future depends only on the present and not on past history.  Suppose for instance we want to know the probability that a student will get an A in his math class the 4^th year of high school and he has completed his 3^rd year. In general, we might expect that this prediction will depend upon what grades he got in his 1^st, 2^nd, and 3^rd years. This would be a Markov process if only the 3^rd (current ) year had any bearing, and that the grades in the previous two years could be ignored.

Membrane is a thin, typically planar structure or material that separates two environments. Because it sits between environments or phases and has a finite volume, it can be referred to as an interphase rather than an interface. Membranes selectively control mass transport between the phases or environments.

The Metropolis Algorithm The Metropolis algorithm is a method to arrive at a stable configuration. Start with a random initial configuration. With each time step, make one change in the configuration. Decide if the change is acceptable. If so, incorporate the change into the configuration. Repeat process until no further major changes result. Deciding what change is acceptable is the key step. The model must have a given probability of change, which can be determined by calculations or estimated by intuition. Then, we generate a random number (a facility provided by most programming software) between 0 and 1. If the random number generator has a uniform distribution between 0 and 1, the probability of a generated random number to be lower than 0.8 is 80%. So, we can compare the random number with the probability of change. If the random number is lower, the change will occur. If the random number is higher, the change is rejected, and the process restarts by proposing another change, comparing a random number with its probability of acceptance, and accepting or rejecting the change. Most accepted changes lead to lower energy configurations. Energy configurations of higher energy are also accepted, although with a lower probability.

Model is a physical, mathematical, or otherwise logical representation of a system, entity, phenomenon, or process.

Monte Carlo is a technique for producing estimates of "true" outcomes of stochastic processes by simply running many iterations of the model process and averaging the outcomes together. Results are given as statistics, eg mean and standard deviation of variable X. Monte Carlo is a very wealthy town in the country of Monaco known for its casino, gambling, beaches, glamour, and sightings of famous people.

N

Normalized eigenvector is a scaled vector whose length is 1.

Null hypothesis is the prediction that an observed difference is due to chance alone and not due to a systematic cause; this hypothesis is tested by statistical analysis, and accepted or rejected.

P

Passive transport is is a means of moving biochemicals, and other atomic or molecular substances, across membranes. Unlike active transport, this process does not involve chemical energy. Passive transport is dependent on the permeability of the cell membrane, which, in turn, is dependent on the organization and characteristics of the membrane lipids and proteins. The four main kind of passive transport are diffusion, facilitated diffusion, filtration and osmosis.

Phase changes in physics refer to melting points of solids to solid crystal shapes; boiling points of liquids; and vaporization of liquids to gases.

Population  is a set of entities concerning which statistical inferences are to be drawn, often based on a random sample taken from the population. For example, if we were interested in generalizations about crows, then we would describe the set of crows that is of interest.

Prediction is a statement or claim that a particular event will occur in the future. To read more about making predictions as a part of the modeling cycle, click here.

Probability is a measure of certainty on a scale of 0 to 1. The chance that an event will occur.

Pseudo-random number is a number belonging to a sequence which appears to be random, but can in fact be generated by a finite computation. They are used extensively in computer science in places where randomness is essential to some application, but cannot be generated due to computation being deterministic.

R

Random means governed by a formal chance process in which the occurrence of previous events is of no value in predicting future events. The probability of assignment of, for example, a given subject to a specified treatment group is fixed and constant (typically 0.50) but the subject's actual assignment cannot be known until it occurs.

Random numbers are a sequence of integers or group of numbers (often in the form of a table) which show absolutely no relationship to each other anywhere in the sequence. At any point, all integers have an equal chance of occurring, and they occur in an unpredictable fashion.

Random number generator is a piece of software that use a system clock and an algorithm to generate a random number within a given range.

Random walk is a method of simulating random movements by beginning at a point and proceeding to a succession of new points by choosing angles at random and moving a standard distance in the directions indicated by these angles.

Recursion means definition of an operation in terms of itself.

S

Sample space is a collection of all possible outcomes. For example, imagine rolling a six sided die. The sample space is {1, 2, 3, 4, 5, 6}.

Skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable. Roughly speaking, a distribution has positive skew (right-skewed) if the higher tail is longer and negative skew (left-skewed) if the lower tail is longer (confusing the two is a common error).

Statistics is the science and practice of developing knowledge through the use of empirical data expressed in quantitative form.

Stochastic is random or probabilistic but with some "direction". For example, the arrival of people at a post office might be random but average properties (such as the queue length) can be predicted.

State In physics and mathematics, the term state is used in several related senses, each of which expresses something about the way a system is.

State vector  is an array containing an object's properties. For example, properties may include object's location, orientation, size, color, etc.

Steady state A state is called stationary if once in this state there is no further change. This can be written as Pnx=x. In the Markov chain example, a stationary state was eventually reached. If we had started with that state, no change would have been seen. This is a dynamic equilibrium not a static one. This means that individuals could move (in the first example from miniature golf to Frisbee golf and vice versa) but that the distribution ( the number of people at each place) remains the same.

Sample is a statistically-significant subset selected and analyzed to estimate the characteristics of a larger group or population; a set of individuals within an organization assessed to provide information on the preferences, opinions, attitudes, and practices of the group they represent.

Spin refers to the angular momentum intrinsic to a body, as opposed to orbital angular momentum, which is the motion of its center of mass about an external point. In classical mechanics, the spin angular momentum of a body is associated with the rotation of the body around its own center of mass. For example, the spin of the Earth is associated with its daily rotation about the polar axis. On the other hand, the orbital angular momentum of the Earth is associated with its annual motion around the Sun.

T

The Transition Matrix P is the matrix of all these probabilities, where x and y take on all possible values. Note that since these are probabilities, all entries of the matrix are numbers ≥0 and ≤1, and that columns add up to 1. The n-step transition matrix Pⁿ is the matrix whose entries are pⁿ(yx), which is the conditional probability Prob(X_t+n=y|X_t=x). The properties of conditional probability and the Markov property lead to a set of relationships called the Chapman-Kolmogorov equations. These in turn simplify Pⁿ so that it is just the product of P n times.

The transition state is the arrangement of atoms present at the highest energy point between reactants and products; also called the "activated complex"