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Modelising the translation errors by suitable mathematical operators in the crystal basis model of the genetic code and requiring that codons prone to be misread encode the same amino-acid, the main features of the organisation in multiplets of the genetic code are described.
A modified equation, the S-K equation, fits data that the current neural conduction equation, the K-R equation, does not. The S-K equation is a modified Heaviside equation, based on a new interpretation of cross terms. Elements of neural anatomy and function are reviewed to put the S-K equation into context. The fit between S-K and resonance-like neural data is then shown. Appendix 1: Derivation of crossterms that represent combinations of physical laws for a line conductor of finite length. Appendix 2: Evaluation of crossterms that represent combinations of physical laws according to consistency arguments. Appendix 3: Some background on resonance. Appendix 4: Web access to some brain modeling, correspondence with NATURE, and discussion of the work in George Johnson's New York Times forums.
It has been proposed that the degeneracy of the genetic code,i.e., the phenomenon that different codons (base triplets) of DNA are transcribed into the same amino acid, may be interpreted as the result of a symmetry breaking process. In the initial work of Hornos & Hornos this picture was developed in the framework of simple Lie algebras. Here, we explore the possibility of explaining the degeneracy of the genetic code using basic classical Lie superalgebras, whose representation theory is sufficiently well understood, at least as far as typical representations are concerned. In the present paper, we give the complete list of all typical codon representations (typical 64 -dimensional irreducible representations), whereas in the second part, we shall present the corresponding branching rules and discuss which of them reproduce the multiplet structure of the genetic code.
A sharp TH1/TH2 dichotomy has often been used to define the effects of cytokines on autoimmune diseases. However contradictory results in recent research indicate that the situation may be more complex. We build here a simple mathematical model aimed at settling the contradictions. The model is based on a neural network paradigm, and is applied using Partial Differential Equations (PDE). We show here that a TH1/TH2 paradigm is only an external view of a complex multivariate system.
Following previous models describing a quasi steady state (QSS) for the evolution of HIV infection and AIDS, we have developed a larger formalism simulating the long-term evolution of the QSS We show that the long-term evolution of AIDS cannot be explained by the destruction alone of CD4 T cells, either directly or indirectly. The destruction of CD4 T cells can lead only to a QSS with a lower concentration of CD4 T cells, but CD4 destruction cannot generate the sustained long-term decrease in T cells leading to AIDS. We here suggest some workable explanations.
An SIRS epidemiological model for a vertically transmitted disease is discussed. We give a complete global analysis in terms of three explicit threshold parameters which respectively govern the existence and stability of an endemic proportion equilibrium, the increase of the total population and the growth of the infective population. This paper gereralize the results of Busenberg and van den Driessche.
In this paper, we give a complete analysis of an SIS epidemiological model in a population of varying size with two dissimilar groups of infective individuals. It is mainly based on the discussion of the existence and stability of equilibria of the proportions system and the result is in terms of a threshold parameter which governs the stability of the disease free equilibrium.
An SIRS type model of disease transmission in an open environment is discussed. We use the Poincaré index together with a perturbation method to show that the endemic proportions need not be unique.
In the near future, all the human genes will be identified. But understanding the functions coded in the genes is a much harder problem. For example, by using block entropy, one has that the DNA code is closer to a random code then written text, which in turn is less ordered then an ordinary computer code; see \cite{schmitt}. Instead of saying that the DNA is badly written, using our programming standards, we might say that it is written in a different style -- an evolutionary style. We will suggest a way to search for such a style in a quantified manner by using an artificial life program, and by giving a definition of general codes and a definition of style for such codes.
Finding a basis/coordinate system that can efficiently represent an input data stream by viewing them as realizations of a stochastic process is of tremendous importance in many fields including data compression and computational neuroscience. Two popular measures of such efficiency of a basis are sparsity (measured by the expected $\ell^p$ norm, $0 < p \leq 1$) and statistical independence (measured by the mutual information). Gaining deeper understanding of their intricate relationship, however, remains elusive. Therefore, we chose to study a simple synthetic stochastic process called the spike process, which puts a unit impulse at a random location in an $n$-dimensional vector for each realization. For this process, we obtained the following results: 1) The standard basis is the best both in terms of sparsity and statistical independence if $n \geq 5$ and the search of basis is restricted within all possible orthonormal bases in $R^n$; 2) If we extend our basis search in all possible invertible linear transformations in $R^n$, then the best basis in statistical independence differs from the one in sparsity; 3) In either of the above, the best basis in statistical independence is not unique, and there even exist those which make the inputs completely dense; 4) There is no linear invertible transformation that achieves the true statistical independence for $n > 2$.
We describe in a rudimentary fashion how SVM(support vector machine) plays the role of classifier in a mathematical setting. We then discuss its application in the study of multiple SNP(single nucleotide polymorphism) variations. Also presented is a set of preliminary test results with clinical data.
A basis under which a given set of realizations of a stochastic process can be represented most sparsely (the so-called best sparsifying basis (BSB)) and the one under which such a set becomes as less statistically dependent as possible (the so-called least statistically-dependent basis (LSDB)) are important for data compression and have generated interests among computational neuroscientists as well as applied mathematicians. Here we consider these bases for a particularly simple stochastic process called ``generalized spike process'', which puts a single spike--whose amplitude is sampled from the standard normal distribution--at a random location in the zero vector of length $\ndim$ for each realization. Unlike the ``simple spike process'' which we dealt with in our previous paper and whose amplitude is constant, we need to consider the kurtosis-maximizing basis (KMB) instead of the LSDB due to the difficulty of evaluating differential entropy and mutual information of the generalized spike process. By computing the marginal densities and moments, we prove that: 1) the BSB and the KMB selects the standard basis if we restrict our basis search within all possible orthonormal bases in ${\mathbb R}^n$; 2) if we extend our basis search to all possible volume-preserving invertible linear transformations, then the BSB exists and is again the standard basis whereas the KMB does not exist. Thus, the KMB is rather sensitive to the orthonormality of the transformations under consideration whereas the BSB is insensitive to that. Our results once again support the preference of the BSB over the LSDB/KMB for data compression applications as our previous work did.
I present three models of plant--pathogen interactions. The models are stochastic and spatially explicit at the scale of individual plants. For each model, I use a version of pair approximation or moment closure along with a separation of timescales argument to determine the effects of spatial clustering on threshold structure. By computing the spatial structure early in an invasion, I find explicit corrections to mean field theory. In the first chapter, I present a lattice model of a disease that is not directly lethal to its host, but affects its ability to compete with neighbors. I use a type of pair approximation to determine conditions for invasions and coexistence. In the second chapter, I study a basic SIR epidemic point process in continuous space. I implement a multiplicative moment closure scheme to compute the threshold transmission rate as a function of spatial parameters. In the final chapter, I model the evolution of pathogen resistance when two plant species share a pathogen. Evolution may lead to non--resistance by a host that finds the disease to be a useful weapon. I use a lattice model with the ordinary pair approximation assumption to study phenotypic evolution via repeated invasions by novel strains.
The present paper studies a feedback regulation problem, which may be interpreted as an adaptive control problem, but has not yet been studied in the control literature. The problem, which arises in at least two different biological applications (Hopf bifurcations in the auditory system, and neural integrators used to maintain persistent neural activity), is that of tuning a parameter so as to bring it to a value at which a bifurcation takes place.
We study a mathematical model of a blood circulation network which is a generalization of the coronary model proposed by Smith, Pullan and Hunter. We prove the existence and uniqueness of the solution to the initial-boundary value problem and discuss the continuity of dependence of the solution and its derivatives on initial, boundary and forcing functions and their derivatives.
A hallmark of a plethora of intracellular signaling pathways is the spatial separation of activation and deactivation processes that potentially results in precipitous gradients of activated proteins. The classical Metabolic Control Analysis (MCA), which quantifies the influence of an individual process on a system variable as the control coefficient, cannot be applied to spatially separated protein networks. The present paper unravels the principles that govern the control over the fluxes and intermediate concentrations in spatially heterogeneous reaction networks. Our main results are two types of the control summation theorems. The first type is a non-trivial generalization of the classical theorems to systems with spatially and temporally varying concentrations. In this generalization, the process of diffusion, which enters as the result of spatial concentration gradients, plays a role similar to other processes such as chemical reactions and membrane transport. The second summation theorem is completely novel. It states that the control by the membrane transport, the diffusion control coefficient multiplied by two, and a newly introduced control coefficient associated with changes in the spatial size of a system (e.g., cell), all add up to one and zero for the control over flux and concentration. Using a simple example of a kinase/phosphatase system in a spherical cell, we speculate that unless active mechanisms of intracellular transport are involved, the threshold cell size is limited by the diffusion control, when it is beginning to exceed the spatial control coefficient significantly.
This talk will review a little over a decade's research on applying certain stochastic models to biological sequence analysis. The models themselves have a longer history, going back over 30 years, although many novel variants have arisen since that time. The function of the models in biological sequence analysis is to summarize the information concerning what is known as a motif or a domain in bioinformatics, and to provide a tool for discovering instances of that motif or domain in a separate sequence segment. We will introduce the motif models in stages, beginning from very simple, non-stochastic versions, progressively becoming more complex, until we reach modern profile HMMs for motifs. A second example will come from gene finding using sequence data from one or two species, where generalized HMMs or generalized pair HMMs have proved to be very effective.
The nervous system displays a variety of rhythms in both waking and sleep. These rhythms have been closely associated with different behavioral and cognitive states, but it is still unknown how the nervous system makes use of these rhythms to perform functionally important tasks. To address those questions, it is first useful to understood in a mechanistic way the origin of the rhythms, their interactions, the signals which create the transitions among rhythms, and the ways in which rhythms filter the signals to a network of neurons. This talk discusses how dynamical systems have been used to investigate the origin, properties and interactions of rhythms in the nervous system. It focuses on how the underlying physiology of the cells and synapses of the networks shape the dynamics of the network in different contexts, allowing the variety of dynamical behaviors to be displayed by the same network. The work is presented using a series of related case studies on different rhythms. These case studies are chosen to highlight mathematical issues, and suggest further mathematical work to be done. The topics include: different roles of excitation and inhibition in creating synchronous assemblies of cells, different kinds of building blocks for neural oscillations, and transitions among rhythms. The mathematical issues include reduction of large networks to low dimensional maps, role of noise, global bifurcations, use of probabilistic formulations.