Dynamical landscapes of cell fate decisions.

The generation of cellular diversity during development involves differentiating cells transitioning between discrete cell states. In the 1940s, the developmental biologist Conrad Waddington introduced a landscape metaphor to describe this process. The developmental path of a cell was pictured as a ball rolling through a terrain of branching valleys with cell fate decisions represented by the branch points at which the ball decides between one of two available valleys. Here we discuss progress in constructing quantitative dynamical models inspired by this view of cellular differentiation. We describe a framework based on catastrophe theory and dynamical systems methods that provides the foundations for quantitative geometric models of cellular differentiation. These models can be fit to experimental data and used to make quantitative predictions about cellular differentiation. The theory indicates that cell fate decisions can be described by a small number of decision structures, such that there are only two distinct ways in which cells make a binary choice between one of two fates. We discuss the biological relevance of these mechanisms and suggest the approach is broadly applicable for the quantitative analysis of differentiation dynamics and for determining principles of developmental decisions.

Temperature regulates NF-κB dynamics and function through timing of A20 transcription.

NF-κB signaling plays a pivotal role in control of the inflammatory response. We investigated how the dynamics and function of NF-κB were affected by temperature within the mammalian physiological range (34 °C to 40 °C). An increase in temperature led to an increase in NF-κB nuclear/cytoplasmic oscillation frequency following Tumor Necrosis Factor alpha (TNFα) stimulation. Mathematical modeling suggested that this temperature sensitivity might be due to an A20-dependent mechanism, and A20 silencing removed the sensitivity to increased temperature. The timing of the early response of a key set of NF-κB target genes showed strong temperature dependence. The cytokine-induced expression of many (but not all) later genes was insensitive to temperature change (suggesting that they might be functionally temperature-compensated). Moreover, a set of temperature- and TNFα-regulated genes were implicated in NF-κB cross-talk with key cell-fate-controlling pathways. In conclusion, NF-κB dynamics and target gene expression are modulated by temperature and can accurately transmit multidimensional information to control inflammation.

Mapping global sensitivity of cellular network dynamics: sensitivity heat maps and a global summation law.

The dynamical systems arising from gene regulatory, signalling and metabolic networks are strongly nonlinear, have high-dimensional state spaces and depend on large numbers of parameters. Understanding the relation between the structure and the function for such systems is a considerable challenge. We need tools to identify key points of regulation, illuminate such issues as robustness and control and aid in the design of experiments. Here, I tackle this by developing new techniques for sensitivity analysis. In particular, I show how to globally analyse the sensitivity of a complex system by means of two new graphical objects: the sensitivity heat map and the parameter sensitivity spectrum. The approach to sensitivity analysis is global in the sense that it studies the variation in the whole of the model's solution rather than focusing on output variables one at a time, as in classical sensitivity analysis. This viewpoint relies on the discovery of local geometric rigidity for such systems, the mathematical insight that makes a practicable approach to such problems feasible for highly complex systems. In addition, we demonstrate a new summation theorem that substantially generalizes previous results for oscillatory and other dynamical phenomena. This theorem can be interpreted as a mathematical law stating the need for a balance between fragility and robustness in such systems.

Uncovering the design principles of circadian clocks: mathematical analysis of flexibility and evolutionary goals.

In this paper, we present the mathematical details underlying both an approach to the flexibility of regulatory networks and an analytical characterization of evolutionary goals of circadian clock networks. A fundamental problem in cellular regulation is to understand the relation between the form of regulatory networks and their function. Circadian clocks present a particularly interesting instance of this. Recent work has shown that they have complex structures involving multiple interconnected feedback loops with both positive and negative feedback. We address the question of why they have such a complex structure and argue that it is to provide the flexibility necessary to simultaneously attain multiple key properties of circadian clocks such as robust entrainment and temperature compensation. To do this we address two fundamental problems: (A) to understand the relationships between the key evolutionary aims of the clock and (B) to ascertain how flexible the clock's structure is. To address the first problem we use infinitesimal response curves (IRCs), a tool that we believe will be of general utility in the analysis of regulatory networks. To understand the second problem we introduce the flexibility dimension d, show how to calculate it and then use it to analyse a range of models. We believe our results will generalize to a broad range of regulatory networks.

Modelling antigenic drift in weekly flu incidence.

Since influenza in humans is a major public health threat, the understanding of its dynamics and evolution, and improved prediction of its epidemics are important aims. Underlying its multi-strain structure is the evolutionary process of antigenic drift whereby epitope mutations give mutant virions a selective advantage. While there is substantial understanding of the molecular mechanisms of antigenic drift, until now there has been no quantitative analysis of this process at the population level. The aim of this study is to develop a predictive model that is of a modest-enough structure to be fitted to time series data on weekly flu incidence. We observe that the rate of antigenic drift is highly non-uniform and identify several years where there have been antigenic surges where a new strain substantially increases infective pressure. The SIR-S approach adopted here can also be shown to improve forecasting in comparison to conventional methods.

Design principles underlying circadian clocks.

A fundamental problem for regulatory networks is to understand the relation between form and function: to uncover the underlying design principles of the network. Circadian clocks present a particularly interesting instance, as recent work has shown that they have complex structures involving multiple interconnected feedback loops with both positive and negative feedback. While several authors have speculated on the reasons for this, a convincing explanation is still lacking. We analyse both the flexibility of clock networks and the relationships between various desirable properties such as robust entrainment, temperature compensation, and stability to environmental variations and parameter fluctuations. We use this to argue that the complexity provides the flexibility necessary to simultaneously attain multiple key properties of circadian clocks. As part of our analysis we show how to quantify the key evolutionary aims using infinitesimal response curves, a tool that we believe will be of general utility in the analysis of regulatory networks. Our results suggest that regulatory and signalling networks might be much less flexible and of lower dimension than their apparent complexity would suggest.

Quantifying the strength of ligand antagonism in TCR triggering.

Triggering of the T cell receptor (TCR) may be antagonized by ligands that are slight variants of the immunogenic peptide. This paper proposes a mathematical model to quantify the strength of the antagonistic effect. The model is based on the kinetics of association and dissociation of TCR and peptide/major histocompatibility (pMHC) molecules, and incorporates TCR triggering according to a kinetic proofreading mechanism. Model analysis indicates that while the average lifetime of the TCR/pMHC complex is the basic determinant of the contribution to TCR triggering made by the ligand, the affinity of the ligand and its MHC presentation level are also important. However, these contributions depend on the kinetic limitation regime. There is a continuum of limitation regimes, at the extremes of which are found TCR limitation and MHC limitation. Both ligand affinity and TCR and pMHC densities determine whether TCR triggering is TCR limited or MHC limited. The changing importance of affinity and antigen presentation level under various kinetic limitation regimes may explain the respective roles of antagonistic and agonistic self peptides in thymic selection. Moreover, TCR down-regulation under TCR-limited conditions may allow the T cell to differentiate between the average lifetime of the TCR/pMHC complex and the presentation level of the ligand. A method for experimental differentiation between passive and active antagonistic effects is proposed which exploits the differences between TCR and MHC limitation.

A reliable and safe T cell repertoire based on low-affinity T cell receptors.

Antigens are presented to T cells as short peptides bound to MHC molecules on the surface of body cells. The binding between MHC/peptides and T cell receptors (TCRs) has a low affinity and is highly degenerate. Nevertheless, TCR-MHC/peptide recognition results in T cell activation of high specificity. Moreover, the immune system is able to mount a cellular response when only a small fraction of the MHC molecules on an antigen-presenting cell is occupied by foreign peptides, while autoimmunity remains relatively rare. We consider how to reconcile these seemingly contradictory facts using a quantitative model of TCR signalling and T cell activation. Taking into account the statistics of TCR recognition and antigen presentation, we show that thymic selection can produce a working T cell repertoire which will produce safe and effective responses, that is, recognizes foreign antigen presented at physiological levels while tolerating self. We introduce "activation curves" as a useful tool to study the repertoire's statistical activation properties.

A moment closure model for sexually transmitted disease transmission through a concurrent partnership network.

A moment closure model of sexually transmitted disease spread through a concurrent partnership network is developed. The model employs pair approximations of higher-order correlations to derive equations of motion in terms of numbers of pairs and singletons. The model is derived from an underlying stochastic process of partnership network formation and disease transmission. The model is analysed numerically; and the final size and time evolution are considered for various levels of concurrency, as measured by the concurrency index kappa3 of Kretzschmar and Morris. Additionally, a new way of calculating R0 for spatial network models is developed. It is found that concurrency significantly increases R0 and the final size of a sexually transmitted disease, with some interesting exceptions.

Markov chain Monte Carlo analysis of human Y-chromosome microsatellites provides evidence of biased mutation.

We describe a Markov Chain Monte Carlo analysis of five human Y- chromosome microsatellite polymorphisms based on samples from five diverse populations. Our analysis provides strong evidence for mutational bias favoring increase in length at all loci. Estimates of population coalescent times and population size from our two largest samples, one African and one European, suggest that the African population is older but smaller and that the English East Anglian population has undergone significant expansion, being larger but younger. We conclude that Markov Chain Monte Carlo analysis of microsatellite haplotypes can uncover information not apparent when the microsatellites are considered independently. Incorporation of population size as a variable should allow us to estimate the timing and magnitude of major historical population trends.

Dynamics of T-cell antagonism: enhanced viral diversity and survival.

In rapidly evolving viruses the detection of virally infected cells can possibly be subverted by the production of altered peptides. There are peptides with single amino acid changes that can dramatically change T-cell responses, e.g. a loss of cytotoxic activity. They are still recognized by the T cell, but the signals required for effector function are only partially delivered. Thus, altered peptide presenting cells can act as decoy targets for specific immune responses. The existence of altered peptides in vivo has been demonstrated in hepatitis B and HIV. Using a mathematical model we address the question of how these altered peptides can affect the virus-immune system dynamics, and demonstrate that virus survival is enhanced. If the mutation rate of the virus is sufficient, one observes complex dynamics in which the antagonism acts so as to maintain the viral diversity, possibly leading to the development of a mutually antagonistic network or a continual turnover of escape mutants. In either case the pathogen is able to outrun the immune system. Indeed, sometimes the enhancement is so great that a virus that would normally be cleared by the immune system is able to outrun it.

Correlation models for childhood epidemics.

One of the simplest set of equations for the description of epidemics (the SEIR equations) has been much studied, and produces reasonable approximations to the dynamics of communicable disease. However, it has long been recognized that spatial and social structure are important if we are to understand the long-term persistence and detailed behaviour of disease. We will introduce three pair models which attempt to capture the underlying heterogeneous structure by studying the connections and correlations between individuals. Although modelling the correlations necessarily leads to more complex equations, this pair formulation naturally incorporates the local dynamical behaviour generating more realistic persistence. In common with other studies on childhood diseases we will focus our attention on measles, for which the case returns are particularly well documented and long running.

Dynamics and evolution: evolutionarily stable attractors, invasion exponents and phenotype dynamics.

We extend the ideas of evolutionary dynamics and stability to a very broad class of biological and other dynamical systems. We simultaneously develop the general mathematical theory and a discussion of some illustrative examples. After developing an appropriate formulation for the dynamics, we define the notion of an evolutionary stable attractor (ESA) and give some samples of ESAS with simple and complex dynamics. We discuss the relationship between our theory and that for ESSS in classical linear evolutionary game theory by considering some dynamical extensions. We then introduce and develop our main mathematical tool, the invasion exponent. This allows analytical and numerical analysis of relatively complex situations, such as the coevolution of multiple species with chaotic population dynamics. Using this, we introduce the notion of differential selective pressure which for generic systems is nonlinear and characterizes internal ESAS. We use this to analytically determine the ESAS in our previous examples. Then we introduce the phenotype dynamics which describe how a population with a distribution of phenotypes changes in time with or without mutations. We discuss the relation between the asymptotic states of this and the ESAS. Finally, we use our mathematical formulation to analyse a non-reproductive form of evolution in which various learning rules compete and evolve. We give a very tentative economic application which has interesting ESAS and phenotype dynamics.

Detecting chaos in a noisy time series.

We propose a new method for detecting low-dimensional chaotic time series when there is dynamical noise present. The method identifies the sign of the largest Liapunov exponent and thus the presence or absence of chaos. It also shows when it is possible to assign a value to the exponent. This approach can work for short time series of only 500 points. We analyse several real time series including chickenpox and measles data from New York City. For model systems it correctly identifies important spatial scales at which noise and nonlinear effects are important. We propose a further technique for estimating the level of noise in real time series if it is difficult to detect by the former method.

Chaotic stochasticity: a ubiquitous source of unpredictability in epidemics.

We address the question of whether or not childhood epidemics such as measles and chickenpox are chaotic, and argue that the best explanation of the observed unpredictability is that it is a manifestation of what we call chaotic stochasticity. Such chaos is driven and made permanent by the fluctuations from the mean field encountered in epidemics, or by extrinsic stochastic noise, and is dependent upon the existence of chaotic repellors in the mean field dynamics. Its existence is also a consequence of the near extinctions in the epidemic. For such systems, chaotic stochasticity is likely to be far more ubiquitous than the presence of deterministic chaotic attractors. It is likely to be a common phenomenon in biological dynamics.