Descriptive Modeling of the Dynamical Systems and Determination of Feedback Homeostasis at Different Levels of Life Organization

The state-of-art research in the field of life’s organization confronts the need to investigate a number of interacting components, their properties and conditions of sustainable behaviour within a natural system. In biology, ecology and life sciences, the performance of such stable system is usually related to homeostasis, a property of the system to actively regulate its state within a certain allowable limits. In our previous work, we proposed a deterministic model for systems’ homeostasis. The model was based on dynamical system’s theory and pairwise relationships of competition, amensalism and antagonism taken from theoretical biology and ecology. However, the present paper proposes a different dimension to our previous results based on the same model. In this paper, we introduce the influence of inter-component relationships in a system, wherein the impact is characterized by direction (neutral, positive, or negative) as well as its (absolute) value, or strength. This makes the model stochastic which, in our opinion, is more consistent with real-world elements affected by various random factors. The case study includes two examples from areas of hydrobiology and medicine. The models acquired for these cases enabled us to propose a convincing explanation for corresponding phenomena identified by different types of natural systems.     

Modelling atypical CYP3A4 kinetics: principles and pragmatism

The Michaelis-Menten model, and the existence of a single active site for the interaction of substrate with drug metabolizing enzyme, adequately describes a substantial number of in vitro metabolite kinetic data sets for both clearance and inhibition determination. However, in an increasing number of cases (involving most notably, but not exclusively, CYP3A4), atypical kinetic features are observed, e.g., auto- and heteroactivation; partial, cooperative, and substrate inhibition; concentration-dependent effector responses (activation/inhibition); limited substrate substitution and inhibitory reciprocity necessitating sub-group classification. The phenomena listed above cannot be readily interpreted using single active site models and the literature indicates that three types of approaches have been adopted. First the 'nai ve' approach of using the Michaelis-Menten model regardless of the kinetic behaviour, second the 'empirical' approach (e.g., employing the Hill or uncompetitive inhibition equations to model homotropic phenomena of sigmoidicity and substrate inhibition, respectively) and finally, the 'mechanistic' approach. The later includes multisite kinetic models derived using the same rapid equilibrium/steady-state assumptions as the single-site model. These models indicate that 2 or 3 binding sites exist for a given CYP3A4 substrate and/or effector. Multisite kinetic models share common features, depending on the substrate kinetics and the nature of the effector response observed in vitro, which allow a generic model to be proposed. Thus although more complex than the other two approaches, they show more utility and can be comprehensively applied in relatively simple versions that can be readily generated from generic model. Multisite kinetic features, observed in isolated hepatocytes as well as in microsomes from hepatic tissue and heterologous expression systems, may be evident in substrate depletion-time profiles as well as in metabolite formation rates. Failure to adequately account for multisite kinetic phenomena will compromise any attempts to predict human drug clearance and drug-drug interaction potential from in vitro data.     

A model of adaptation based on relaxation phenomena in the neural membrane

Summary Adaptation in neural systems is due to a number of phenomena which include characteristic receptor response, delayed inhibition and other network properties, as well as adaptation on the single unit level. The last effect has been isolated and studied extensively by a number of experimenters. We are proposing a model which accounts for this effect. It is assumed that the post-spike hyperpolarizations are cumulative and are caused by two distinct membrane processes of differing relaxation times. The model involves a number of parameters which can be fitted to existing neurophysiological data.     

Some thoughts on the requirements for a theory of biology

Complexity and stability are the ubiquitous characteristics of all biological phenomena. The theoretical suggestions presented here acknowledge this and attempt to capitalize on the property of selectiveness that complexity induces. It is from this that a belief in the existence of selection rules emerges. What is required of a viable theory is a linguistic means of maneuvering known information so that these selection or limitation rules may be obtained, and thence, by their manipulation, to generate the properties exhibited by biological systems.     

A two-component model for counts of infectious diseases

We propose a stochastic model for the analysis of time series of disease counts as collected in typical surveillance systems on notifiable infectious diseases. The model is based on a Poisson or negative binomial observation model with two components: a parameter-driven component relates the disease incidence to latent parameters describing endemic seasonal patterns, which are typical for infectious disease surveillance data. An observation-driven or epidemic component is modeled with an autoregression on the number of cases at the previous time points. The autoregressive parameter is allowed to change over time according to a Bayesian changepoint model with unknown number of changepoints. Parameter estimates are obtained through the Bayesian model averaging using Markov chain Monte Carlo techniques. We illustrate our approach through analysis of simulated data and real notification data obtained from the German infectious disease surveillance system, administered by the Robert Koch Institute in Berlin. Software to fit the proposed model can be obtained from http://www.statistik.lmu.de/ approximately mhofmann/twins.     

From intracellular signaling to population oscillations: bridging size- and time-scales in collective behavior

Collective behavior in cellular populations is coordinated by biochemical signaling networks within individual cells. Connecting the dynamics of these intracellular networks to the population phenomena they control poses a considerable challenge because of network complexity and our limited knowledge of kinetic parameters. However, from physical systems, we know that behavioral changes in the individual constituents of a collectively behaving system occur in a limited number of well-defined classes, and these can be described using simple models. Here, we apply such an approach to the emergence of collective oscillations in cellular populations of the social amoeba Dictyostelium discoideum. Through direct tests of our model with quantitative in vivo measurements of single-cell and population signaling dynamics, we show how a simple model can effectively describe a complex molecular signaling network at multiple size and temporal scales. The model predicts novel noise-driven single-cell and population-level signaling phenomena that we then experimentally observe. Our results suggest that like physical systems, collective behavior in biology may be universal and described using simple mathematical models.     

Identifying human cells capable of metabolizing various classes of carcinogens

Human cells that appear capable of metabolizing various classes of carcinogens have been identified using one of two methods: metabolism of tritiated benzo(a)pyrene to aqueous-acetone soluble forms or inhibition of cellular DNA synthesis. Each of the assay systems was optimized and the results on 15 human epithelial cell lines were compared. One or more cell lines were found to activate each of four classes of carcinogens examined: polycyclic hydrocarbons, aromatic amines, heterocyclic hydrocarbons, and nitrosamines. Cells that appeared capable of metabolizing polycyclic hydrocarbons or aromatic amines by these methods were also found to produce metabolites which were cytotoxic to cocultivated human xeroderma pigmentosum fibroblasts after a 48-hr exposure to the carcinogen.     

Putative contribution of glucose kinase from Bacillus subtilis to carbon catabolite repression (CCR): a link between enzymatic regulation and CCR?

Carbon catabolite repression (CCR) in Bacillus subtilis is mainly mediated via the central component CcpA. Nevertheless, it has been reported that some sugar metabolizing systems underlie additional CcpA-independent CCR mechanisms. Here, we present evidence supporting a potential function of glucose kinase (GlcK) in CCR, analyzing knockout mutants in CcpA and GlcK. On the one hand, GlcK inactivation has no effect on CCR by glucose or fructose of the xylose system. On the other hand, CCR of the trehalose system is affected but a remaining glucose CCR still persists. This remaining glucose repression can be explained by the appearance of unphosphorylated glucose obtained from cytoplasmic trehalose 6-phosphate hydrolysis, which is detectable in cell culture supernatant.     

Stabilization of inhomogeneous patterns in a diffusion-reaction system under structural and parametric uncertainties

Many phenomena such as neuron firing in the brain, the travelling waves which produce the heartbeat, arrythmia and fibrillation in the heart, catalytic reactions or cellular organization activities, among others, can be described by a unifying paradigm based on a class of nonlinear reaction-diffusion mechanisms. The FitzHugh-Nagumo (FHN) model is a simplified version of such class which is known to capture most of the qualitative dynamic features found in the spatiotemporal signals. In this paper, we take advantage of the dissipative nature of diffusion-reaction systems and results in finite dimensional nonlinear control theory to develop a class of nonlinear feedback controllers which is able to ensure stabilization of moving fronts for the FHN system, despite structural or parametric uncertainty. In the context of heart or neuron activity, this class of control laws is expected to prevent cardiac or neurological disorders connected with spatiotemporal wave disruptions. In the same way, biochemical or cellular organization related with certain functional aspects of life could also be influenced or controlled by the same feedback logic. The stability and robustness properties of the controller will be proved theoretically and illustrated on simulation experiments.     

The mathematical theory of lateral impedance of visual activity and an application of Markov processes

The mathematical relationship describing recurrent lateral inhibition is expressed as a linear operator equation. Under quite general conditions, the operator equation is shown to have a unique nonnegative solution. It is also shown that the linear operator for recurrent impedance is representable as an integral operator and that, when in application to physiological models it is interpreted as recurrent inhibition, the corresponding linear equation assumes a form more general than the well known Hartline-Ratliff equation. Finally, we introduce a class of impedance operators based on the probabilistic theory of Markov processes, solve the corresponding linear integral equation, and apply the theoretical properties of the solution to the analysis of physiological and psychophysical phenomena.     

Simulating Spinal Border Cells and Cerebellar Granule Cells under Locomotion – A Case Study of Spinocerebellar Information Processing

The spinocerebellar systems are essential for the brain in the performance of coordinated movements, but our knowledge about the spinocerebellar interactions is very limited. Recently, several crucial pieces of information have been acquired for the spinal border cell (SBC) component of the ventral spinocerebellar tract (VSCT), as well as the effects of SBC mossy fiber activation in granule cells of the cerebellar cortex. SBCs receive monosynaptic input from the reticulospinal tract (RST), which is an important driving system under locomotion, and disynaptic inhibition from Ib muscle afferents. The patterns of activity of RST neurons and Ib afferents under locomotion are known. The activity of VSCT neurons under fictive locomotion, i.e. without sensory feedback, is also known, but there is little information on how these neurons behave under actual locomotion and for cerebellar granule cells receiving SBC input this is completely unknown. But the available information makes it possible to simulate the interactions between the spinal and cerebellar neuronal circuitries with a relatively large set of biological constraints. Using a model of the various neuronal elements and the network they compose, we simulated the modulation of the SBCs and their target granule cells under locomotion and hence generated testable predictions of their general pattern of modulation under this condition. This particular system offers a unique opportunity to simulate these interactions with a limited number of assumptions, which helps making the model biologically plausible. Similar principles of information processing may be expected to apply to all spinocerebellar systems.     

Reconciling emergences: An information-theoretic approach to identify causal emergence in multivariate data

The broad concept of emergence is instrumental in various of the most challenging open scientific questions—yet, few quantitative theories of what constitutes emergent phenomena have been proposed. This article introduces a formal theory of causal emergence in multivariate systems, which studies the relationship between the dynamics of parts of a system and macroscopic features of interest. Our theory provides a quantitative definition of downward causation, and introduces a complementary modality of emergent behaviour—which we refer to as causal decoupling. Moreover, the theory allows practical criteria that can be efficiently calculated in large systems, making our framework applicable in a range of scenarios of practical interest. We illustrate our findings in a number of case studies, including Conway’s Game of Life, Reynolds’ flocking model, and neural activity as measured by electrocorticography.     

Computational simulation of vibrationally resolved spectra for spin‐forbidden transitions

In this computational study, we illustrate a method for computing phosphorescence and circularly polarized phosphorescence spectra of molecular systems, which takes into account vibronic effects including both Franck‐Condon and Herzberg‐Teller contributions. The singlet and triplet states involved in the phosphorescent emission are described within the harmonic approximation, and the method fully takes mode‐mixing effects into account when evaluating Franck‐Condon integrals. Spin‐orbit couplings, which are responsible for these otherwise forbidden phenomena, are accounted for by means of a relativistic two‐component time‐dependent density functional theory method. The model is applied to two types of chiral systems: camphorquinone, a rigid organic system that allows for an extensive benchmark, and some members of a class of iridium complexes. The merits and shortcomings of the methods are discussed, and some perspectives for future developments are offered.     

Stochastic models for an enzyme reaction in an open linear system

A stochastic model is developed for an enzyme reaction in an open linear system. The proposed model assumes that the open system maintains the concentration of substrate and inhibitor at constant levels and that the product molecules are removed from the system by a first order reaction. Stochastic models for several enzyme reactions occurring in this open system are shown to correspond to special cases of theGI/M/∞ queue. Takács’ (1958) results for this queueing system are used to obtain the stochastic properties of the enzyme systems. A specific model we studied assumed completely competitive inhibition in an open system. The stationary distribution for the number of product molecules in the system is obtained. The enzyme reaction which incorporated the “intermediate chain hypothesis” can also be investigated by the queueing theory approach. It is shown that for this open system, if the model which incorporated the intermediate chain hypothesis has the same deterministic properties as the Michaelis-Menten model, then the latter has greater stochastic variation than the former. Research supported by the NIH Training Grant No. GM 1237-05 awarded to the Department of Statistics, The Johns Hopkins University. This paper in whole or in part may be reproduced for any purpose of the United States Government.     

A computational model for cell differentiation

In this paper, we present a word set generating mechanism, called cell-differentiation system, inspired by the tissue process formation in multicellular organisms, which might model some properties of evolving communities of living cells at the syntactical level. The tools utilized to model these biological phenomena belong to the formal language theory. In this context chromosomal mutations are defined as operations on strings and the differentiation according to the control of gene expression is represented by some random-context conditions in formal languages.In the presented formal framework we prove that in a simplified form of this formalism, with only one cell-type which is regular, one single cell and no mitosis involved, the problem of establishing whether or not the set of vectors of integers indicating the number of cells in each population, is finite, linear or semilinear, is recursively undecidable. However, one can algorithmically decide whether or not a cell-differentiation system of finite cell-type can produce a specific generation of cells.     

Modeling adaptive biological systems

During the evolution of many systems found in nature, both the system composition and the interactions between components will vary. Equating the dimension with the number of different components, a system which adds or deletes components belongs to a class of dynamical systems with a finite dimensional phase space of variable dimension. We present two models of biochemical systems with a variable phase space, a model of autocatalytic reaction networks in the prebiotic soup and a model of the idiotypic network of the immune system. Each model contains characteristic meta-dynamical rules for constructing equations of motion from component properties. The simulation of each model occurs on two levels. On one level, the equations of motion are integrated to determine the state of each component. On a second level, algorithms which approximate physical processes in the real system are employed to change the equations of motion. Models with meta-dynamical rules possess several advantages for the study of evolving systems. First, there are no explicit fitness functions to determine how the components of the model rank in terms of survivability. The success of any component is a function of its relationship to the rest of the system. A second advantage is that since the phase space representation of the system is always finite but continually changing, we can explore a potentially infinite phase space which would otherwise be inaccessible with finite computer resources. Third, the enlarged capacity of systems with meta-dynamics for variation allows us to conduct true evolution experiments. The modeling methods presented here can be applied to many real biological systems. In the two studies we present, we are investigating two apparent properties of adaptive networks. With the simulation of the prebiotic soup, we are most interested in how a chemical reaction network might emerge from an initial state of relative disorder. With the study of the immune system, we study the self-regulation of the network including its ability to distinguish between species which are part of the network and those which are not.     

On control and optimal control in biodynamic systems

In any control system for which the number of independent controls is smaller than the number of degrees of freedom to be controlled, our choice of control in any state is restricted to a submanifold of smaller dimension than the tangent space. This simple fact has a number of important consequences for questions of biological import; we consider its implications for adaptation, for senescent phenomena and for the determination of tertiary structures of polypeptides through control of certain average properties. We also formulate the Pontryagin Maximum Principle of Optimal control theory in such a way as to inquire whether specific biodynamic systems can be regarded as optimal with respect to rate of accumulation of particular quantities of the system. We find that if this is possible, the quantity in question must play the role of a clock.     

Multiscale analysis of pattern formation via intercellular signalling

Lateral inhibition, a juxtacrine signalling mechanism by which a cell adopting a particular fate inhibits neighbouring cells from doing likewise, has been shown to be a robust mechanism for the formation of fine-grained spatial patterns (in which adjacent cells in developing tissues diverge to achieve contrasting states of differentiation), provided that there is sufficiently strong feedback. The fine-grained nature of these patterns poses problems for analysis via traditional continuum methods since these require that significant variation takes place only over lengthscales much larger than an individual cell and such systems have therefore been investigated primarily using discrete methods. Here, however, we apply a multiscale method to derive systematically a continuum model from the discrete Delta-Notch signalling model of Collier et al. (J.R. Collier, N.A.M. Monk, P.K. Maini, J.H. Lewis, Pattern formation by lateral inhibition with feedback: a mathematical model of Delta-Notch intercellular signalling, J. Theor. Biol., 183, 1996, 429-446) under particular assumptions on the parameters, which we use to analyse the generation of fine-grained patterns. We show that, on the macroscale, the contact-dependent juxtacrine signalling interaction manifests itself as linear diffusion, motivating the use of reaction-diffusion-based models for such cell-signalling systems. We also analyse the travelling-wave behaviour of our system, obtaining good quantitative agreement with the discrete system.     

Loss of stability-a new model for stress relaxation in plant cell walls

This study addresses the mechanism of wall stress relaxation in growing plant cells. The current viscoelastic model of cell wall relaxation, which dates from the work of Preston, Cleland, Lockhart, and others in the 1960s, has serious shortcomings. It has been shown however that the theory of loss of stability (LOS) can be applied to materials in tension, leading to the conclusion that the relaxation of stresses in the walls of any pressure vessel is rigorously modeled using LOS. We propose that LOS also provides a more appropriate and versatile model of stress relaxation in growing plant cells. We argue that when treated as a manifestation of LOS, the regulation of cell turgor has a rigorous and demonstrable basis in the geometrical and physical properties of the cell wall and the cell's ability to import water. Thus plant cell growth can be regarded as an inherently self-limiting process, tunable by biochemical or structural means. Lastly, despite the current limitations of our model, we apply direct measurement of elastic modulus, wall thickness and cell radius obtained from cylindrical Chara corallina cells to generate an initial calculation of critical pressures in a hypothetical spherical cell with the same material properties.     

Contact-inhibited chemotaxis in de novo and sprouting blood-vessel growth

Blood vessels form either when dispersed endothelial cells (the cells lining the inner walls of fully formed blood vessels) organize into a vessel network (vasculogenesis), or by sprouting or splitting of existing blood vessels (angiogenesis). Although they are closely related biologically, no current model explains both phenomena with a single biophysical mechanism. Most computational models describe sprouting at the level of the blood vessel, ignoring how cell behavior drives branch splitting during sprouting. We present a cell-based, Glazier-Graner-Hogeweg model (also called Cellular Potts Model) simulation of the initial patterning before the vascular cords form lumens, based on plausible behaviors of endothelial cells. The endothelial cells secrete a chemoattractant, which attracts other endothelial cells. As in the classic Keller-Segel model, chemotaxis by itself causes cells to aggregate into isolated clusters. However, including experimentally observed VE-cadherin-mediated contact inhibition of chemotaxis in the simulation causes randomly distributed cells to organize into networks and cell aggregates to sprout, reproducing aspects of both de novo and sprouting blood-vessel growth. We discuss two branching instabilities responsible for our results. Cells at the surfaces of cell clusters attempting to migrate to the centers of the clusters produce a buckling instability. In a model variant that eliminates the surface-normal force, a dissipative mechanism drives sprouting, with the secreted chemical acting both as a chemoattractant and as an inhibitor of pseudopod extension. Both mechanisms would also apply if force transmission through the extracellular matrix rather than chemical signaling mediated cell-cell interactions. The branching instabilities responsible for our results, which result from contact inhibition of chemotaxis, are both generic developmental mechanisms and interesting examples of unusual patterning instabilities.