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Computational approaches to parameter estimation and model selection in immunologyBaker, Christopher T. H.; Bocharov, Gennady; Ford, Judith M.; Lumb, Patricia M.; Norton, Stewart J.; Paul, C. A. H.; Junt, Tobias; Krebs, Philippe; Ludewig, Burkhard (Elsevier, 20051201)This article seeks to illustrate the computational implementation of an informationtheoretic approach (associated with a maximum likelihood treatment) to modelling in immunology.

Computational aspects of timelag models of Marchuk type that arise in immunologyBaker, Christopher T. H.; Bocharov, Gennady; University of Chester ; Institute of Numerical Mathematics, Russian Academy of Sciences (de Gryuter, 2005)In his book published in English translation in 1983, Marchuk proposed a set of evolutionary equations incorporating delaydifferential equations, and the corresponding initial conditions as a model ('Marchuk's model') for infectious diseases. The parameters in this model (and its subsequent extensions) represent scientifically meaningful characteristics. For a given infection, the parameters can be estimated using observational data on the course of the infection. Sensitivity analysis is an important tool for understanding a particular model; this can be viewed as an issue of stability with respect to structural perturbations in the model. Examining the sensitivity of the models based on delay differential equations leads to systems of neutral delay differential equations. Below we formulate a general set of equations for the sensitivity coefficients for models comprising neutral delay differential equations. We discuss computational approaches to the sensitivity of solutions — (i) sensitivity to the choice of model, in particular, to the lag parameter τ > 0 and (ii) sensitivity to the initial function — of dynamical systems with time lag and illustrate them by considering the sensitivity of solutions of timelag models of Marchuk type.

Computational modelling with functional differential equations: Identification, selection, and sensitivityBaker, Christopher T. H.; Bocharov, Gennady; Paul, C. A. H.; Rihan, F. A. R.; University College Chester ; Institute of Numerical Mathematics, Russian Academy of Sciences ; University of Salford (Elsevier, 200505)Mathematical models based upon certain types of differential equations, functional differential equations, or systems of such equations, are often employed to represent the dynamics of natural, in particular biological, phenomena. We present some of the principles underlying the choice of a methodology (based on observational data) for the computational identification of, and discrimination between, quantitatively consistent models, using scientifically meaningful parameters. We propose that a computational approach is essential for obtaining meaningful models. For example, it permits the choice of realistic models incorporating a timelag which is entirely natural from the scientific perspective. The timelag is a feature that can permit a close reconciliation between models incorporating computed parameter values and observations. Exploiting the link between information theory, maximum likelihood, and weighted least squares, and with distributional assumptions on the data errors, we may construct an appropriate objective function to be minimized computationally. The minimizer is sought over a set of parameters (which may include the timelag) that define the model. Each evaluation of the objective function requires the computational solution of the parametrized equations defining the model. To select a parametrized model, from amongst a family or hierarchy of possible bestfit models, we are able to employ certain indicators based on informationtheoretic criteria. We can evaluate confidence intervals for the parameters, and a sensitivity analysis provides an expression for an information matrix, and feedback on the covariances of the parameters in relation to the best fit. This gives a firm basis for any simplification of the model (e.g., by omitting a parameter).

Determining control parameters for dendritic cellcytotoxic T lymphocyte interactionLudewig, Burkhard; Krebs, Philippe; Junt, Tobias; Metters, Helen; Ford, Neville J.; Anderson, Roy M.; Bocharov, Gennady; University of Zürich ; University of Zürich ; University of Zürich ; University of Zürich ; University College Chester ; Imperial College, University of London ; Institute of Numerical Mathematics, Russian Academy of Sciences (WILEYVCH Verlag GmbH & Co. KGaA, 20040805)Dendritic cells (DC) are potent immunostimulatory cells facilitating antigen transport to lymphoid tissues and providing efficient stimulation of T cells. A series of experimental studies in mice demonstrated that cytotoxic T lymphocytes (CTL) can be efficiently induced by adoptive transfer of antigenpresenting DC. However, the success of DCbased immunotherapeutic treatment of human cancer, for example, is still limited because the details of the regulation and kinetics of the DCCTL interaction are not yet completely understood. Using a combination of experimental mouse studies, mathematical modeling, and nonlinear parameter estimation, we analyzed the population dynamics of DCinduced CTL responses. The model integrates a predatorpreytype interaction of DC and CTL with the nonlinear compartmental dynamics of T cells. We found that T cell receptor avidity, the halflife of DC, and the rate of CTLmediated DCelimination are the major control parameters for optimal DCinduced CTL responses. For induction of high avidity CTL, the number of adoptively transferred DC was of minor importance once a minimal threshold of approximately 200 cells per spleen had been reached. Taken together, our study indicates that the availability of high avidity T cells in the recipient in combination with the optimal application regimen is of prime importance for successful DCbased immunotherapy.

A geneticalgorithm approach to simulating human immunodeficiency virus evolution reveals the strong impact of multiply infected cells and recombinationBocharov, Gennady; Ford, Neville J.; Edwards, John T.; Breinig, Tanja; WainHobson, Simon; Meyerhans, Andreas; Institute of Numerical Mathematics, Russian Academy of Sciences ; University of Chester ; University of Chester ; University of the Saarland ; Unité de Rétrovirologie Moléculaire, Institut Pasteur ; University of the Saarland (Society for General Microbiology / High Wire Press, 20051101)It has been previously shown that the majority of human immunodeficiency virus type 1 (HIV1)infected splenocytes can harbour multiple, divergent proviruses with a copy number ranging from one to eight. This implies that, besides point mutations, recombination should be considered as an important mechanism in the evolution of HIV within an infected host. To explore in detail the possible contributions of multiinfection and recombination to HIV evolution, the effects of major microscopic parameters of HIV replication (i.e. the pointmutation rate, the crossover number, the recombination rate and the provirus copy number) on macroscopic characteristics (such as the Hamming distance and the abundance of npoint mutants) have been simulated in silico. Simulations predict that multiple provirus copies per infected cell and recombination act in synergy to speed up the development of sequence diversity. Point mutations can be fixed for some time without fitness selection. The time needed for the selection of multiple mutations with increased fitness is highly variable, supporting the view that stochastic processes may contribute substantially to the kinetics of HIV variation in vivo.

Neutral delay differential equations in the modelling of cell growthBaker, Christopher T. H.; Bocharov, Gennady; Rihan, F. A. R.; University of Chester (University of Chester, 2008)In this contribution, we indicate (and illustrate by example) roles that may be played by neutral delay differential equations in modelling of certain cell growth phenomena that display a time lag in reacting to events. We explore, in this connection, questions involving the sensitivity analysis of models and related mathematical theory; we provide some associated numerical results.

On some aspects of casual and neutral equations used in mathematical modellingBaker, Christopher T. H.; Bocharov, Gennady; Parmuzin, Evgeny I.; Rihan, F. A. R.; University of Chester (University of Chester, 2007)The problem that motivates the considerations here is the construction of mathematical models of natural phenomena that depend upon past states. The paper divides naturally into two parts: in the first, we expound the interconnection between ordinary differential equations, delay differential equations, neutral delaydifferential equations and integral equations (with emphasis on certain linear cases). As we show, this leads to a natural hierarchy of model complexity when such equations are used in mathematical and computational modelling, and to the possibility of reformulating problems either to facilitate their numerical solution or to provide mathematical insight, or both. Volterra integral equations include as special cases the others we consider. In the second part, we develop some practical and theoretical consequences of results given in the first part. In particular, we consider various approaches to the definition of an adjoint, we establish (notably, in the context of sensitivity analysis for neutral delaydifferential equations) roles for welldefined adjoints and ‘quasiadjoints’, and we explore relationships between sensitivity analysis, the variation of parameters formulae, the fundamental solution and adjoints.

Recombination: Multiply infected spleen cells in HIV patientsJung, Andreas; Maier, Reinhard; Vartanian, JeanPierre; Bocharov, Gennady; Jung, Volker; Fischer, Ulrike; Meese, Eckart; WainHobson, Simon; Meyerhans, Andreas (Nature Publishing Group, 20020711)The genome of the human immunodeficiency virus is highly prone to recombination, although it is not obvious whether recombinants arise infrequently or whether they are constantly being spawned but escape identification because of the massive and rapid turnover of virus particles. Here we use fluorescence in situ hybridization to estimate the number of proviruses harboured by individual splenocytes from two HIV patients, and determine the extent of recombination by sequencing amplified DNA from these cells. We find an average of three or four proviruses per cell and evidence for huge numbers of recombinants and extensive genetic variation. Although this creates problems for phylogenetic analyses, which ignore recombination effects, the intracellular variation may help to broaden immune recognition.

Underwhelming the immune response: Effect of slow virus growth on CD8+Tlymphocyte responsesBocharov, Gennady; Burkhard, Ludewig; Bertoletti, Antonio; Klenerman, Paul; Junt, Tobias; Krebs, Philippe; Luzyanina, Tatyana; Fraser, Cristophe; Anderson, Roy M.; University of London/Institute of Numerical Mathematics, Russian Academy of Sciences ; University of Zurich ; University College London ; Oxford University ; University of Zurich ; University of Zurich ; Leuven University ; University of London ; University of London (American Society for Microbiology, 20040212)The speed of virus replication has typically been seen as an advantage for a virus in overcoming the ability of the immune system to control its population growth. Under some circumstances, the converse may also be true: more slowly replicating viruses may evoke weaker cellular immune responses and therefore enhance their likelihood of persistence. Using the model of lymphocytic choriomeningitis virus (LCMV) infection in mice, we provide evidence that slowly replicating strains induce weaker cytotoxicTlymphocyte (CTL) responses than a more rapidly replicating strain. Conceptually, we show a "bellshaped" relationship between the LCMV growth rate and the peak CTL response. Quantitative analysis of human hepatitis C virus infections suggests that a reduction in virus growth rate between patients during the incubation period is associated with a spectrum of disease outcomes, from fulminant hepatitis at the highest rate of viral replication through acute resolving to chronic persistence at the lowest rate. A mathematical model for virusCTL population dynamics (analogous to predator [CTL]prey [virus] interactions) is applied in the clinical datadriven analysis of acute hepatitis B virus infection. The speed of viral replication, through its stimulus of host CTL responses, represents an important factor influencing the pathogenesis and duration of virus persistence within the human host. Viruses with lower growth rates may persist in the host because they "sneak through" immune surveillance.