Some of David Rand's current papers

On this page I list some of my recent papers.

Blue titles: Systems Biology/Mathematical Biology. Black titles: Dynamical Systems

O. E. Akman, J.C.W. Locke, S. Tang, I. Carré, A. J. Millar & D. A. Rand, Global temperature compensation and extended functionality for the Neurospora crassa circadian clock. Submitted

Abstract. A striking and defining feature of circadian clocks, which contrasts with most other oscillators, is the almost constancy of the rhythmic period over a physiological range of temperatures. This is known as temperature compensation. Moreover, given that many rate constants have a Q10 of around two it is remarkable that such clocks remain rhythmic over the physiological temperature range. We introduce a new mathematical model for the Neurospora crassa circadian network which includes recent experimental findings including the discovery that temperature changes affect posttranscriptional regulation by altering the balance of translation between a short and long form of a key protein. This model and a previous one are used to discuss the temperature compensation and the functional temperature range of the Neurospora crassa circadian network and certain mutant forms. The model reproduces all the key experimental data on temperature dependence and arrhythmicity in the mutant and wild-type strains. We present a mechanism for temperature compensation that utilises the presence of two temperature-dependent loops, suggesting a relatively simple mechanism by which temperature compensation could have evolved. We argue that this regulatory structure and the associated tuning that produces temperature compensation also increases the temperature range where the clock is rhythmic.

D A Rand, Mapping the global sensitivity of cellular network dynamics. Submitted.

Abstract. 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 structure and function for such systems is a considerable challenge. We need tools to identify key points of regulation, to illuminate such issues as robustness and control and to aid in the design of experiments. Here I tackle this by developing new techniques for sensitivity analysis, parameter reduction and experimental design and optimisation. In particular, I show how to globally analyse the sensitivity of a complex system by means of a sensitivity heat map; furthermore I provide a method to predict which experimental protocols and perturbations best reveal which aspects of the system and to provide a cost-benefit analysis of the available options for data collection. 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 leads to 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.

Heron E., Finkenstadt B., Rand D. (2007) Statistical inference for delayed transcriptional gene regulation, an application to the Hes1 system, working paper to be submitted.
(with A. A Pinto) Solenoid Functions for Hyperbolic Sets. To appear in the Katok Birthday Volume.
(with A. A Pinto and F Ferreira) Hausdorff dimension bounds for smoothness of holonomies for codimension 1 hyperbolic dynamics. J. Diff. Eqns (to appear)
(with A. A. Pinto) Geometric measures for hyperbolic surface dynamics. 40pp Submitted

Abstract. Using new concepts that we call a measure solenoid function and cocyle-gap pairs we present a moduli space for all hyperbolic systems on surfaces that have an invariant measure that is absolutely continuous with respect to Hausdorff measure. For Anosov systems such a moduli space was constructed by Cawley, but to extend the result to non-Anosov systems one is forced to adopt a different approach to deal with C1+ structures of leaves with a Cantor set structure. We extend Sinai’s eigenvalue formula for C1+-Hausdorff realisations of Anosov diffeomorphisms to hyperbolic diffeomorphisms. We characterise the Lipschitz conjugacy of such hyperbolic systems. classes in a number of ways, for example, in terms of eigenvalues of periodic points and Gibbs measures.

(with B.V. Shulgin, D. Salazar & A. J. Millar) Uncovering the design principles of circadian clocks: Mathematical analysis of flexibility and evolutionary goals. Journal of Theoretical Biology, 238(3) (2006) 616-635.
(with H van den Berg) Quantitating T cell responsiveness. Download PDF Immunological Reviews. In press.
(with B. V. Shulgin, D. Salazar & A. J. Millar) Design principles underlying circadian clocks. Journal of The Royal Society, Interface 1 (2004) online at DOI: 10.1098/rsif.2004.0014 PDF Supplementary Info.
Abstract. For regulatory networks a fundamental problem 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 of this. Recent work has shown that they have complex structures involving multiple interconnected feedback loops. This has led several authors to speculate on the reasons for this but a convincing explanation is still lacking. We argue that the complexity is to provide the flexibility necessary to ensure multiple key properties of circadian clocks, simultaneously, such as temperature compensation and robust entrainment. As part of our analysis we show how to quantify the key evolutionary aims using infinitesimal response curves (IRCs), a tool that we believe will be of general utility in the analysis of regulatory networks. Our results suggest that regulatory and signaling networks might be much less flexible and lower-dimensional than is suggested by their apparent complexity.
(with B F Finkenstädt & A Morton) Measuring antigenic drift and surge in Influenza: statistical inference on immunity loss. Statistics in Medicine. 24(22) (2005) 3447-3461
Abstract. Since Influenza in humans is a major public health threat causing significant mortality and morbidity, the prediction of Influenza epidemics is an important aim. It is made substantially more difficult by antigenic drift and shift. In this paper we develop a pair of simple stochastic SIR-S epidemic models to address this problem. The first model incorporates constant loss of immunity due to antigenic drift while the second extends this to allow for time-dependent shifts in the level of immunity. An ansatz for maximum likelihood estimation is presented and the model parameters are estimated from weekly surveillance. One of the conclusions of this first estimation of the epidemiological parameters is the observation that the rate of antigenic drift is highly non-uniform and in several years we observe an antigenic surge where many immunes to suddenly move into the susceptible class.
(with A J Millar and B Shulgin) Bifurcation analysis of phase bistability in circadian clocks forced by skeleton photoperiod.
Abstract. In this paper we study the dynamics of circadian clocks when they are forced by skeleton light-dark cycles. We show how to construct the skeleton light-dark cycles so as to produce interesting dynamics which gives rise to testable experimental signatures that probe the molecular structure. We show to predict these effects using published data.
Correlations and the invasion of cooperation in the Prisoner’s Dilemma. 4pp.
Games with correlations. 35pp.
Abstract. In these two papers I consider interaction structures determined by a general class of networks with variable coordination number such as spatial networks, telecommunications networks and the World-Wide Web. I present a general theory of games that takes such correlations into account. I show that one can model a much broader class of games than before and also give analytical results where previously there were only simulations. I analyse the evolution of cooperation in the Prisoner’s Dilemma game and explain the important simulation results that are in the literature.
(with F. Ferreira & A. A. Pinto) Hausdorff dimension bounds smoothness of holonomies for codimension one hyperbolic attractors.
Abstract. We show that codimension one hyperbolic attractors cannot admit affine structures and therefore that their stable holonomies cannot be more smooth than C1,Hds (where Hds is the Hausdorff dimensions of the unstable leaves).
(with H van den Berg) Foreigness as a matter of degree: The relative immunogenicity of peptide/MHC ligands. J. Theor. Biol. 231 (2004) 535-548.
(with H A van den Berg) Dynamics of T cell activation threshold tuning. J. Theor. Biol. 228 (2004) 397-416. PDF
(with A. A. Pinto) Rigidity of hyperbolic surface dynamics. J. Lond. Math. Soc. 71 (2005) 481-502. PDF
Abstract. This contains a proof of the following result: if the stable and unstable holonomies of a codimension one hyperbolic basic set are respectively C1,HDs and C1,HDu (where Hds and Hsu are respectively the Hausdorff dimensions of the stable and unstable leaves) then for some 0<a<1 the hyperbolic set is C1 a conjugate to an affine model. C1,HD is a little smoother that the Holder class C1 HD.
(with H van den Berg) Antigen presentation as a diversity filter that enhances immune efficacy. J. Theor. Biol. 224(2) 2003 249-267.
Abstract. We analyse the question of how MHC peptide presentation of antigen on T cells could be optimized so as to obtain an effective and safe immune response. We deduce the need for both MHC restriction and high presentation selectivity and find that the optimal selectivity is such that about one pathogen-derived peptide is presented per MHC isoform, on the average. We also indicate upper and lower bounds to the number of MHC isoforms per individual based on detectability requirements. Thus we deduce that an important role of MHC presentation is to act as a filter that limits the diversity of antigen presentation.
(with A. A. Pinto) Teichmüller spaces and HR structures for hyperbolic surface dynamics. Ergodic Theory and Dynamical Systems 22(6) (2002) 1905-1931.
(with H van den Berg and N J Burroughs) Quantifying the strength of ligand antagonism in TCR triggering. Bull. Math. Biol. 64 (2002) 781-808.
(with H. A. van den Berg & N. J. Burroughs), A reliable and safe T cell repertoire based on low-affinity receptors. J. Theor. Biol. 209 (2001) 465-486.
(with A. A. Pinto) Smoothness of holonomies for codimension 1 hyperbolic dynamics. Bull. London Mathematical Society 34 (2002) 341-352
(with A. A. Pinto) Existence, uniqueness and ratio decomposition for Gibbs states via duality. Ergodic Theory & Dynamical Systems 21 (2) (2001) 533-544.
(with C Bauch) A Moment Closure Model for STD Transmission Through a Concurrent Network Partnership. Proc. R. Soc. series B. 269 (2000) 2019-2027.
(with G. Cooper N. Burroughs, D. C. Rubinstein and W. Amos) Markov Chain Monte Carlo analysis of human Y chromosome microsatellites provides evidence of biased mutation. PNAS 96 (1999) 11916-11921. (Available at http://intl.pnas.org/cgi/reprint/96/21/11916.pdf.)
(with Mutimer, D., Pillay, D., Dragon, E., Tang, H., Ahmed, M., O'Donnell, K., Shaw, J., Burroughs, N.J., Rand, D., Cane, P., Martin, B., Buchan, S., Boxall, E., Barmat, S., Gutekunst, K., McMaster, P. & Elias, E) High pre-treatment serum hepatitis B virus titre predicts failure of lamivudine prophylaxis and graft re-infection after liver transplantation. J. Hepatology 30, (1999) 715-721.
(with M. van Baalen) The evolution of altruism in viscous populations. J. Theor. Biol. 193 (1998) 631-648.
(with N. Burroughs, D Mutimer, E. Elias & D.Pillay), The consequences of quasi-species heterogeneity and within-host ecology for anti-viral therapy in HBV. In Therapies for Viral Hepatitis (ed. Schinazi, Sommadossi & Thomas) International Medical Press (1998) pp. 335-343.
(with N. J. Burroughs) Complex dynamics of cytoxic T-cell antagonism enhances virus survival. Proc. R. Soc. B 265 (1998) 529-535
Correlation equations and pair approximations for spatial ecologies. In Advanced Ecological Theory. Principles and Applications (ed. J. McGlade), pp. 100-142, Blackwell Science, 1999

Address:
Mathematics Institute
University of Warwick
Coventry CV4 7AL
United Kingdom