|
Cellular Potts Model
In computational biology, a Cellular Potts model (CPM, also known as the Glazier-Graner-Hogeweg model) is a computational model of cells and tissues. It is used to simulate individual and collective cell behavior, tissue morphogenesis and cancer development. CPM describes cells as deformable objects with a certain volume, that can adhere to each other and to the medium in which they live. The formalism can be extended to include cell behaviours such as cell migration, growth and division, and cell signalling. The first CPM was proposed for the simulation of cell sorting by François Graner and James Glazier as a modification of a large-Q Potts model. CPM was then popularized by Paulien Hogeweg for studying morphogenesis. Although the model was developed to describe biological cells, it can also be used to model individual parts of a biological cell, or even regions of fluid. Model description The CPM consists of a rectangular Euclidean lattice, where each cell is a subset of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computational Biology
Computational biology refers to the use of data analysis, mathematical modeling and computational simulations to understand biological systems and relationships. An intersection of computer science, biology, and big data, the field also has foundations in applied mathematics, chemistry, and genetics. It differs from biological computing, a subfield of computer engineering which uses bioengineering to build computers. History Bioinformatics, the analysis of informatics processes in biological systems, began in the early 1970s. At this time, research in artificial intelligence was using network models of the human brain in order to generate new algorithms. This use of biological data pushed biological researchers to use computers to evaluate and compare large data sets in their own field. By 1982, researchers shared information via punch cards. The amount of data grew exponentially by the end of the 1980s, requiring new computational methods for quickly interpreting ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclidean Space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any positive integer dimension (mathematics), dimension, including the three-dimensional space and the ''Euclidean plane'' (dimension two). The qualifier "Euclidean" is used to distinguish Euclidean spaces from other spaces that were later considered in physics and modern mathematics. Ancient History of geometry#Greek geometry, Greek geometers introduced Euclidean space for modeling the physical space. Their work was collected by the Greek mathematics, ancient Greek mathematician Euclid in his ''Elements'', with the great innovation of ''mathematical proof, proving'' all properties of the space as theorems, by starting from a few fundamental properties, called ''postulates'', which either were considered as eviden ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stochastic Cellular Automata
Stochastic cellular automata or probabilistic cellular automata (PCA) or random cellular automata or locally interacting Markov chains are an important extension of cellular automaton. Cellular automata are a discrete-time dynamical system of interacting entities, whose state is discrete. The state of the collection of entities is updated at each discrete time according to some simple homogeneous rule. All entities' states are updated in parallel or synchronously. Stochastic Cellular Automata are CA whose updating rule is a stochastic one, which means the new entities' states are chosen according to some probability distributions. It is a discrete-time random dynamical system. From the spatial interaction between the entities, despite the simplicity of the updating rules, complex behaviour may emerge like self-organization. As mathematical object, it may be considered in the framework of stochastic processes as an interacting particle system in discrete-time. See for a more deta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
CompuCell3D
CompuCell3D (CC3D) is a three-dimensional C++ and Python software problem solving environment for simulations of biocomplexity problems, integrating multiple mathematical orphogenesismodels. These include the cellular Potts model (CPM) which can model cell clustering, growth, division, death, adhesion, and volume and surface area constraints; as well as partial differential equation solvers for modeling reaction–diffusion of external chemical fields and cell type automata for differentiation. By integrating these models CompuCell3D enables modeling of cellular reactions to external chemical fields such as secretion or resorption, and responses such as chemotaxis and haptotaxis. CompuCell3D is conducive for experimentation and testing biological models by providing a flexible and extensible package, with many different levels of control. High-level steering is possible through CompuCell Player, an interactive GUI built upon Qt threads which execute in parallel with the computat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chemokine
Chemokines (), or chemotactic cytokines, are a family of small cytokines or signaling proteins secreted by cells that induce directional movement of leukocytes, as well as other cell types, including endothelial and epithelial cells. In addition to playing a major role in the activation of host immune responses, chemokines are important for biological processes, including morphogenesis and wound healing, as well as in the pathogenesis of diseases like cancers. Cytokine proteins are classified as chemokines according to behavior and structural characteristics. In addition to being known for mediating chemotaxis, chemokines are all approximately 8-10 kilodaltons in mass and have four cysteine residues in conserved locations that are key to forming their 3-dimensional shape. These proteins have historically been known under several other names including the ''SIS family of cytokines'', ''SIG family of cytokines'', ''SCY family of cytokines'', ''Platelet factor-4 superfamily'' or ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Haptotaxis
Haptotaxis (from Greek ἅπτω (hapto, "touch, fasten") and τάξις (taxis, "arrangement, order")) is the directional motility or outgrowth of cells, e.g. in the case of axonal outgrowth, usually up a gradient of cellular adhesion sites or substrate-bound chemoattractants (the gradient of the chemoattractant being expressed or bound on a surface, in contrast to the classical model of chemotaxis, in which the gradient develops in a soluble fluid.). These gradients are naturally present in the extracellular matrix (ECM) of the body during processes such as angiogenesis or artificially present in biomaterials where gradients are established by altering the concentration of adhesion sites on a polymer substrate. Clinical Significance Haptotaxis plays a major role in the efficient healing of wounds. For example, when corneal integrity is compromised, epithelial cells quickly cover the damaged area by proliferation and migration (haptotaxis). In the corneal stroma, keratocytes wit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chemotaxis
Chemotaxis (from '' chemo-'' + ''taxis'') is the movement of an organism or entity in response to a chemical stimulus. Somatic cells, bacteria, and other single-cell or multicellular organisms direct their movements according to certain chemicals in their environment. This is important for bacteria to find food (e.g., glucose) by swimming toward the highest concentration of food molecules, or to flee from poisons (e.g., phenol). In multicellular organisms, chemotaxis is critical to early development (e.g., movement of sperm towards the egg during fertilization) and development (e.g., migration of neurons or lymphocytes) as well as in normal function and health (e.g., migration of leukocytes during injury or infection). In addition, it has been recognized that mechanisms that allow chemotaxis in animals can be subverted during cancer metastasis. The aberrant chemotaxis of leukocytes and lymphocytes also contribute to inflammatory diseases such as atherosclerosis, asthma, and arthr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lagrange Multiplier
In mathematical optimization, the method of Lagrange multipliers is a strategy for finding the local maxima and minima of a function subject to equality constraints (i.e., subject to the condition that one or more equations have to be satisfied exactly by the chosen values of the variables). It is named after the mathematician Joseph-Louis Lagrange. The basic idea is to convert a constrained problem into a form such that the derivative test of an unconstrained problem can still be applied. The relationship between the gradient of the function and gradients of the constraints rather naturally leads to a reformulation of the original problem, known as the Lagrangian function. The method can be summarized as follows: in order to find the maximum or minimum of a function f(x) subjected to the equality constraint g(x) = 0, form the Lagrangian function :\mathcal(x, \lambda) = f(x) + \lambda g(x) and find the stationary points of \mathcal considered as a function of x and the Lagrange mu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kronecker Delta
In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise: \delta_ = \begin 0 &\text i \neq j, \\ 1 &\text i=j. \end or with use of Iverson brackets: \delta_ = =j, where the Kronecker delta is a piecewise function of variables and . For example, , whereas . The Kronecker delta appears naturally in many areas of mathematics, physics and engineering, as a means of compactly expressing its definition above. In linear algebra, the identity matrix has entries equal to the Kronecker delta: I_ = \delta_ where and take the values , and the inner product of vectors can be written as \mathbf\cdot\mathbf = \sum_^n a_\delta_b_ = \sum_^n a_ b_. Here the Euclidean vectors are defined as -tuples: \mathbf = (a_1, a_2, \dots, a_n) and \mathbf= (b_1, b_2, ..., b_n) and the last step is obtained by using the values of the Kronecker delta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monte Carlo Method In Statistical Physics
Monte Carlo in statistical physics refers to the application of the Monte Carlo method to problems in statistical physics, or statistical mechanics. Overview The general motivation to use the Monte Carlo method in statistical physics is to evaluate a multivariable integral. The typical problem begins with a system for which the Hamiltonian is known, it is at a given temperature and it follows the Boltzmann statistics. To obtain the mean value of some macroscopic variable, say A, the general approach is to compute, over all the phase space, PS for simplicity, the mean value of A using the Boltzmann distribution: :\langle A\rangle=\int_ A_ \frac d\vec. where E(\vec)=E_ is the energy of the system for a given state defined by \vec - a vector with all the degrees of freedom (for instance, for a mechanical system, \vec = \left(\vec, \vec \right) ), \beta\equiv 1/k_bT and :Z= \int_ e^d\vec is the partition function. One possible approach to solve this multivariable integral is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Metropolis Algorithm
A metropolis () is a large city or conurbation which is a significant economic, political, and cultural center for a country or region, and an important hub for regional or international connections, commerce, and communications. A big city belonging to a larger urban agglomeration, but which is not the core of that agglomeration, is not generally considered a metropolis but a part of it. The plural of the word is ''metropolises'', although the Latin plural is ''metropoles'', from the Greek ''metropoleis'' (). For urban centers outside metropolitan areas that generate a similar attraction on a smaller scale for their region, the concept of the regiopolis ("regio" for short) was introduced by urban and regional planning researchers in Germany in 2006. Etymology Metropolis (μητρόπολις) is a Greek word, coming from μήτηρ, ''mḗtēr'' meaning "mother" and πόλις, ''pólis'' meaning "city" or "town", which is how the Greek colonies of antiquity referred to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Principle Of Minimum Energy
The principle of minimum energy is essentially a restatement of the second law of thermodynamics. It states that for a closed system, with constant external parameters and entropy, the internal energy will decrease and approach a minimum value at equilibrium. External parameters generally means the volume, but may include other parameters which are specified externally, such as a constant magnetic field. In contrast, for isolated systems (and fixed external parameters), the second law states that the entropy will increase to a maximum value at equilibrium. An isolated system has a fixed total energy and mass. A closed system, on the other hand, is a system which is connected to another, and cannot exchange matter (i.e. particles), but can transfer other forms of energy (e.g. heat), to or from the other system. If, rather than an isolated system, we have a closed system, in which the entropy rather than the energy remains constant, then it follows from the first and second laws of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
.jpg "Click for more on -> Computational Biology")
