Perspectives on mathematical modeling for receptor-mediated processes

Mathematical modeling is a potent in silico tool that can help investigate, interpret, and predict the behavior of biological systems. The first step is to develop a working hypothesis of the biology. Then by "translating" the biological phenomena into equations, models can harness the power of mathematical analysis techniques to explore the dynamics and interactions of the biological components. Models can be used together with traditional experimental models to help design new experiments, test hypotheses, identify mechanisms, and predict outcomes. This article reviews the process of building, calibrating, and using mathematical models in the context of the kinetics of receptor and signal transduction biology. An example model related to the androgen receptor-mediated regulation of the prostate is presented to illustrate the steps in the modeling process and to highlight the potential for mathematical modeling in this area.     

Monitoring sequence space as a test for the target of selection in viruses

An essential feature of viral quasispecies, predicted from quasispecies theory, is that the target of selection is the mutant distribution as a whole. To test molecularly the mutant composition selected from a viral quasispecies we reconstructed a mutant distribution using 19 antigenic variants of foot-and-mouth disease virus (FMDV). Each variant was marked by a specific amino acid replacement at a major antigenic site of the virus that conferred resistance to a monoclonal antibody (mAb). The variants were introduced in the mutant spectrum of a biological FMDV clone, at a frequency commonly found in FMDV quasispecies. The reconstructed quasispecies (and a number of control populations) were allowed to replicate in the presence or absence of the mAb. The mutant distribution that became dominant as a result of antibody selection included at least ten of the 19 mutants initially used to reconstruct the quasispecies. No such biased mutant repertoire was found in control populations. The results show that a mutant distribution was selected, and are incompatible with selection of an individual genome, which then generated multiple mutants upon further replication. An ample representation of variants immediately following a selection event should contribute to subsequent adaptability of the virus.     

Spatial stability analysis of Eigen's quasispecies model and the less than five membered hypercycle under global population regulation

A generalization of the “constant overall organization” constraint of Eigen's quasispecies and hypercycle models, called herein “global population regulation”, is shown to lead to mathematically tractable spatial generalizations of these two models. The spatially uniform steady state of Eigen's quasispecies model is shown to be stable and globally attracting for all possible values of the mutation and replication rates. In contrast, the spatially and temporally uniform solutions to the hypercycle with fewer than five members, the only ones insensitive to stochastic perturbations, are shown to be unstable, and a lower bound to the spatial inhomogeneities is obtained. The prospect that the spatially localized hypercycle might be immune to various instabilities cited in the literature is then briefly considered. Although spatial localization makes possible a much richer dynamical repertoire than previously considered, it is also more difficult to understand how Darwinian selection of hypercycles could result in a unique genetic code.     

The relationship between the error catastrophe, survival of the flattest, and natural selection

Background  

The quasispecies model is a general model of evolution that is generally applicable to replication up to high mutation rates. It predicts that at a sufficiently high mutation rate, quasispecies with higher mutational robustness can displace quasispecies with higher replicative capacity, a phenomenon called "survival of the flattest". In some fitness landscapes it also predicts the existence of a maximum mutation rate, called the error threshold, beyond which the quasispecies enters into error catastrophe, losing its genetic information. The aim of this paper is to study the relationship between survival of the flattest and the transition to error catastrophe, as well as the connection between these concepts and natural selection.     

Classical and quantum dynamics on p-adic trees of ideas

We propose mathematical models of information processes of unconscious and conscious thinking (based on p-adic number representation of mental spaces). Unconscious thinking is described by classical cognitive mechanics (which generalizes Newton's mechanics). Conscious thinking is described by quantum cognitive mechanics (which generalizes the pilot wave model of quantum mechanics). The information state and motivation of a conscious cognitive system evolve under the action of classical information forces and a new quantum information force, namely, conscious force. Our model might provide mathematical foundations for some cognitive and psychological phenomena: collective conscious behavior, connection between physiological and mental processes in a biological organism, Freud's psychoanalysis, hypnotism, homeopathy. It may be used as the basis of a model of conscious evolution of life.     

Effective degree network disease models

An effective degree approach to modeling the spread of infectious diseases on a network is introduced and applied to a disease that confers no immunity (a Susceptible-Infectious-Susceptible model, abbreviated as SIS) and to a disease that confers permanent immunity (a Susceptible-Infectious-Recovered model, abbreviated as SIR). Each model is formulated as a large system of ordinary differential equations that keeps track of the number of susceptible and infectious neighbors of an individual. From numerical simulations, these effective degree models are found to be in excellent agreement with the corresponding stochastic processes of the network on a random graph, in that they capture the initial exponential growth rates, the endemic equilibrium of an invading disease for the SIS model, and the epidemic peak for the SIR model. For each of these effective degree models, a formula for the disease threshold condition is derived. The threshold parameter for the SIS model is shown to be larger than that derived from percolation theory for a model with the same disease and network parameters, and consequently a disease may be able to invade with lower transmission than predicted by percolation theory. For the SIR model, the threshold condition is equal to that predicted by percolation theory. Thus unlike the classical homogeneous mixing disease models, the SIS and SIR effective degree models have different disease threshold conditions.     

A Mechanistic Investigation of the Algae Growth “Droop” Model

In this work a mechanistic explanation of the classical algae growth model built by M. R. Droop in the late sixties is proposed. We first recall the history of the construction of the "predictive" variable yield Droop model as well as the meaning of the introduced cell quota. We then introduce some theoretical hypotheses on the biological phenomena involved in nutrient storage by the algae that lead us to a "conceptual" model. Though more complex than Droop's one, our model remains accessible to a complete mathematical study: its confrontation to the Droop model shows both have the same asymptotic behavior. However, while Droop's cell quota comes from experimental bio-chemical measurements not related to intra-cellular biological phenomena, its analogous in our model directly follows our theoretical hypotheses. This new model should then be looked at as a re-interpretation of Droop's work from a theoretical biologist's point of view.     

Memory in retroviral quasispecies: experimental evidence and theoretical model for human immunodeficiency virus

Viral quasispecies may possess a molecular memory of their past evolutionary history, imprinted on minority components of the mutant spectrum. Here we report experimental evidence and a theoretical model for memory in retroviral quasispecies in vivo. Apart from replicative memory associated with quasispecies dynamics, retroviruses may harbour a "cellular" or "anatomical" memory derived from their integrative cycle and the presence of viral reservoirs in body compartments. Three independent sets of data exemplify the two kinds of memory in human immunodeficiency virus type 1 (HIV-1). The data provide evidence of re-emergence of sequences that were hidden in cellular or anatomical compartments for extended periods of infection, and recovery of a quasispecies from pre-existing genomes. We develop a three-component model that incorporates the essential features of the quasispecies dynamics of retroviruses exposed to selective pressures. Significantly, a numerical study based on this model is in agreement with the experimental data, further supporting the existence of both replicative and reservoir memory in retroviral quasispecies.     

The concept of a changing receptor concentration: implications for the theory of drug action

Drugs are considered to produce their effects on biological tissues either by altering some physical property of cells or by interacting with specific cellular components, called receptors. Most drugs and endogenous neurotransmitters act on highly selective receptors located on the outer surface membrane of cells. These receptors were believed, until recently, to be stationary on the cell surface and to be present in unvarying numbers. Consequently, most early theorists modeled the drug-receptor interaction on the basis of stationary and static receptor molecules. The substantial advances in our understanding of drug action based on these models have partly justified this view. However, recent electron microscopic studies have revealed the presence of structures, including "coated" pits and vesicles, that appear to provide a mechanism by which cell surface receptors might be internalized in a process of endocytosis. The precise intracellular fate of these internalized receptors is unknown, but based on present understanding, it seems reasonable to believe that some are destroyed intracellularly whereas others are recycled to the cell surface. The importance of such processes to pharmacologic theory is a new awareness of a cellular pathway that is capable of internalizing drugs, receptors, or both. The implications of such a process to the theory of drug action extends to some unexplained drug phenomena such as down regulation, drug tolerance, tachyphyllaxis, and partial agonism. We present herein the theoretical framework for a model of drug action that incorporates the possibility of receptor internalization and subsequent degradation, recycling, or replacement.     

Mathematical Models for Sleep-Wake Dynamics: Comparison of the Two-Process Model and a Mutual Inhibition Neuronal Model

Sleep is essential for the maintenance of the brain and the body, yet many features of sleep are poorly understood and mathematical models are an important tool for probing proposed biological mechanisms. The most well-known mathematical model of sleep regulation, the two-process model, models the sleep-wake cycle by two oscillators: a circadian oscillator and a homeostatic oscillator. An alternative, more recent, model considers the mutual inhibition of sleep promoting neurons and the ascending arousal system regulated by homeostatic and circadian processes. Here we show there are fundamental similarities between these two models. The implications are illustrated with two important sleep-wake phenomena. Firstly, we show that in the two-process model, transitions between different numbers of daily sleep episodes can be classified as grazing bifurcations. This provides the theoretical underpinning for numerical results showing that the sleep patterns of many mammals can be explained by the mutual inhibition model. Secondly, we show that when sleep deprivation disrupts the sleep-wake cycle, ostensibly different measures of sleepiness in the two models are closely related. The demonstration of the mathematical similarities of the two models is valuable because not only does it allow some features of the two-process model to be interpreted physiologically but it also means that knowledge gained from study of the two-process model can be used to inform understanding of the behaviour of the mutual inhibition model. This is important because the mutual inhibition model and its extensions are increasingly being used as a tool to understand a diverse range of sleep-wake phenomena such as the design of optimal shift-patterns, yet the values it uses for parameters associated with the circadian and homeostatic processes are very different from those that have been experimentally measured in the context of the two-process model.     

In silico identification of functional divergence between the multiple groEL gene paralogs in Chlamydiae

Background

RNA molecules, through their dual appearance as sequence and structure, represent a suitable model to study evolutionary properties of quasispecies. The essential ingredient in this model is the differentiation between genotype (molecular sequences which are affected by mutation) and phenotype (molecular structure, affected by selection). This framework allows a quantitative analysis of organizational properties of quasispecies as they adapt to different environments, such as their robustness, the effect of the degeneration of the sequence space, or the adaptation under different mutation rates and the error threshold associated.

Results

We describe and analyze the structural properties of molecular quasispecies adapting to different environments both during the transient time before adaptation takes place and in the asymptotic state, once optimization has occurred. We observe a minimum in the adaptation time at values of the mutation rate relatively far from the phenotypic error threshold. Through the definition of a consensus structure, it is shown that the quasispecies retains relevant structural information in a distributed fashion even above the error threshold. This structural robustness depends on the precise shape of the secondary structure used as target of selection. Experimental results available for natural RNA populations are in qualitative agreement with our observations.

Conclusion

Adaptation time of molecular quasispecies to a given environment is optimized at values of the mutation rate well below the phenotypic error threshold. The optimal value results from a trade-off between diversity generation and fixation of advantageous mutants. The critical value of the mutation rate is a function not only of the sequence length, but also of the specific properties of the environment, in this case the selection pressure and the shape of the secondary structure used as target phenotype. Certain functional motifs of RNA secondary structure that withstand high mutation rates (as the ubiquitous hairpin motif) might appear early in evolution and be actually frozen evolutionary accidents.     

Perspectives on Mathematical Modeling for Receptor-Mediated Processes

Mathematical modeling is a potent in silico tool that can help investigate, interpret, and predict the behavior of biological systems. The first step is to develop a working hypothesis of the biology. Then by “translating” the biological phenomena into equations, models can harness the power of mathematical analysis techniques to explore the dynamics and interactions of the biological components. Models can be used together with traditional experimental models to help design new experiments, test hypotheses, identify mechanisms, and predict outcomes. This article reviews the process of building, calibrating, and using mathematical models in the context of the kinetics of receptor and signal transduction biology. An example model related to the androgen receptor-mediated regulation of the prostate is presented to illustrate the steps in the modeling process and to highlight the potential for mathematical modeling in this area.     

Dependence of biological treatment rate on species composition in activated sludge or biofilm. II: From models to theory

Traditional biological treatment models are "deduced" from formal chemical kinetics or dynamics of pure microorganism cultures growth. The best formal models give reasonable approximations of the biological treatment model with an ecosystem adaptation (ESA model). The model presented here explains some features of the biological treatment mechanism that cannot be described by formal models.     

Unfinished Stories on Viral Quasispecies and Darwinian Views of Evolution

Experimental evidence that RNA virus populations consist of distributions of mutant genomes, termed quasispecies, was first published 31 years ago. This work provided the earliest experimental support for a theory to explain a system that replicated with limited fidelity and to understand the self-organization and adaptability of early life forms on Earth. High mutation rates and quasispecies dynamics of RNA viruses are intimately related to both viral disease and antiviral treatment strategies. Moreover, the quasispecies concept is being applied to other biological systems such as cancer research in which cellular mutant spectra can be also detected. This review addresses some of the unanswered questions regarding viral and theoretical quasispecies concepts as well as more practical aspects concerning resistance to antiviral treatments and pathogenesis.     

Connecting behavioural biologists and psychologists: Clarifying distinctions and suggestions for further work

This article is a reply to the commentaries on our target article, which relates our group's work on simple heuristics to biological research on rules of thumb. Several commentators contrasted both these approaches with behaviour analysis, in which the patterns of behaviour investigated in the laboratory are claimed to be near-universal attributes, rather than specific to particular appropriate environments. We question this universality. For instance, learning phenomena such Pavlovian or operant conditioning have mostly been studied only in a few generalist species that learn easily; in many natural situations the environment hinders learning as an adaptive strategy. Other supposedly general phenomena such as impulsiveness and matching are outcome models, which several different models of simple cognitive processes might explain. We clarify some confusions about optimisation, optima and optimality modelling. Lastly, we say a little more about how heuristics might be selected, learnt and tuned to suit the current environment.     

Idiotypic networks incorporating T-B cell co-operation. The conditions for percolation

Previous work was concerned with symmetric immune networks of idiotypic interactions amongst B cell clones. The behaviour of these networks was contrary to expectations. This was caused by an extensive percolation of idiotypic signals. Idiotypic activation was thus expected to affect almost all (greater than 10(7] B cell clones. We here analyse whether the incorporation of helper T cells (Th) into these B cell models could cause a reduction in the percolation. Empirical work on idiotypic interactions between Th and B cells however, would suggest that two different idiotypic Th models should be developed: (1) a Th which recognises native B cell idiotypes, i.e. a non-MHC-restricted "ThId" model, and (2) a "classical" MHC-restricted helper T cell model. In the ThId model, the Th-B cell interaction is symmetric. A 2-D model of a Th and a B cell clone that interact idiotypically with each other accounts for various equilibria (i.e. one virgin and two immune states). Introduction of antigen does indeed lead to a state switch from the virgin to the immune state; such a system is thus able to "remember" its exposure to antigen. Idiotypic signals do however, percolate in ThId models via these "B-Th-B-Th" pathways: proliferating Th and B cell clones that interact idiotypically, will always activate each other reciprocally. In the MHC-restricted Th model, Th-B interactions are asymmetric. Because the B cell idiotypes are processed and subsequently presented by MHC molecules, the Th receptor and the native B cell receptor are not expected to be complementary. Thus the Th and the B cells are unable to activate each other reciprocally, and a 2-D Th-B cell model cannot account for idiotypic memory. In contrast to the ThId model, idiotypic activation cannot percolate via "B-Th-B-Th" interactions. Due to the assymmetry idiotypic activation stops at the first Th level. A Th clone cannot activate a subsequent B cell clone: if the B cells recognise the Th cells, they see idiotype but get no help; if the Th cells see the B cells, the B cells are helped but see no idiotype. The percolation along "B-B-B" pathways in these two models is next analysed. Two B cells clones, each helped by one Th clone, are connected by a symmetric idiotypic interaction. It turns out that in both models the second (i.e. anti-idiotypic) B cells (B2) never proliferate.(ABSTRACT TRUNCATED AT 400 WORDS)     

Studies of folding and misfolding using simplified models

Computer simulations are as vital to our studies of biological systems as experiments. They bridge and rationalize experimental observations, extend the experimental "field of view", which is often limited to a specific time or length scale, and, most importantly, provide novel insights into biological systems, offering hypotheses about yet-to-be uncovered phenomena. These hypotheses spur further experimental discoveries. Simplified molecular models have a special place in the field of computational biology. Branded as less accurate than all-atom protein models, they have offered what all-atom molecular dynamics simulations could not--the resolution of the length and time scales of biological phenomena. Not only have simplified models proven to be accurate in explaining or reproducing several biological phenomena, they have also offered a novel multiscale computational strategy for accessing a broad range of time and length scales upon integration with traditional all-atom simulations. Recent computer simulations of simplified models have shaken or advanced the established understanding of biological phenomena. It was demonstrated that simplified models can be as accurate as traditional molecular dynamics approaches in identifying native conformations of proteins. Their application to protein structure prediction yielded phenomenal accuracy in recapitulating native protein conformations. New studies that utilize the synergy of simplified protein models with all-atom models and experiments yielded novel insights into complex biological processes, such as protein folding, aggregation and the formation of large protein complexes.     

An analysis of biological random processes via optimized statistical models

The concept of a sampled probability-density vector is defined. It is shown that a relationship may be established between this new estimation means and the random process, expressed by its moment vector. This is a linear transformation using invariant matrices i.e. matrices which are independent of the random process. Thus, in deriving biological probability-density models, instead of using an analytical model, to estimate their parameters and to check the distributional assumptions, a single probability-density vector is computed, subject to some constraints. An optimized statistical model is, thus, obtained, by minimizing a certain loss function, which expresses the inaccuracy of the model. The invariant matrices permitting to obtain the optimized model, starting from the moment vector, are given and the procedure is illustrated by examples. Then, the concept of a parametric probability-density space is defined and it is shown that each of the vectors belonging to this space may express the stationary, ergodic, random process equally well. Some typical constraints in the probability-density space are investigated. It is shown that the normal (Gaussian) law may be regarded as a very strong constraint in the probability-density space, while the integral law, expressing the cumulative distribution function, is a weak one. Between these extreme cases, the large class of the usual constraints are examined, which are determined by the prior knowledge of the process, as well as by some desired model features. Thus, the concept of a constrained probability-density vector is introduced. By using a linear-programming procedure and by observing some peak constraints as well as some slope-sign ones, an optimized model with desired shape is obtained, where a certain value of the variable has a very high probability. This leads to a procedure which enables synaptic models to be derived. In such a model, the constraints in the probability-density space may be regarded as a new expression of the information transmitted in the nervous system. Moreover, the loss function may express the aptitude of the random process to realize a given message. Thus, by using the optimized statistical model concept, probabilistic models with desired features for various biological processes may be obtained in a simple and general manner.