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Stochastic models of lymphocyte proliferation and death.

ABSTRACT: Quantitative understanding of the kinetics of lymphocyte proliferation and death upon activation with an antigen is crucial for elucidating factors determining the magnitude, duration and efficiency of the immune response. Recent advances in quantitative experimental techniques, in particular intracellular labeling and multi-channel flow cytometry, allow one to measure the population structure of proliferating and dying lymphocytes for several generations with high precision. These new experimental techniques require novel quantitative methods of analysis. We review several recent mathematical approaches used to describe and analyze cell proliferation data. Using a rigorous mathematical framework, we show that two commonly used models that are based on the theories of age-structured cell populations and of branching processes, are mathematically identical. We provide several simple analytical solutions for a model in which the distribution of inter-division times follows a gamma distribution and show that this model can fit both simulated and experimental data. We also show that the estimates of some critical kinetic parameters, such as the average inter-division time, obtained by fitting models to data may depend on the assumed distribution of inter-division times, highlighting the challenges in quantitative understanding of cell kinetics.

SUBMITTER: Zilman A

PROVIDER: S-EPMC2948000 | biostudies-literature | 2010 Sep

REPOSITORIES: biostudies-literature

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Stochastic models of lymphocyte proliferation and death.

Zilman Anton A Ganusov Vitaly V VV Perelson Alan S AS

PloS one 20100930 9

Quantitative understanding of the kinetics of lymphocyte proliferation and death upon activation with an antigen is crucial for elucidating factors determining the magnitude, duration and efficiency of the immune response. Recent advances in quantitative experimental techniques, in particular intracellular labeling and multi-channel flow cytometry, allow one to measure the population structure of proliferating and dying lymphocytes for several generations with high precision. These new experimen ...[more]

PMID: 20941358

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Project description:Adaptive immune responses are complex dynamic processes whereby B and T cells undergo division and differentiation triggered by pathogenic stimuli. Deregulation of the response can lead to severe consequences for the host organism ranging from immune deficiencies to autoimmunity. Tracking cell division and differentiation by flow cytometry using fluorescent probes is a major method for measuring progression of lymphocyte responses, both in vitro and in vivo. In turn, mathematical modeling of cell numbers derived from such measurements has led to significant biological discoveries, and plays an increasingly important role in lymphocyte research. Fitting an appropriate parameterized model to such data is the goal of these studies but significant challenges are presented by the variability in measurements. This variation results from the sum of experimental noise and intrinsic probabilistic differences in cells and is difficult to characterize analytically. Current model fitting methods adopt different simplifying assumptions to describe the distribution of such measurements and these assumptions have not been tested directly. To help inform the choice and application of appropriate methods of model fitting to such data we studied the errors associated with flow cytometry measurements from a wide variety of experiments. We found that the mean and variance of the noise were related by a power law with an exponent between 1.3 and 1.8 for different datasets. This violated the assumptions inherent to commonly used least squares, linear variance scaling and log-transformation based methods. As a result of these findings we propose a new measurement model that we justify both theoretically, from the maximum entropy standpoint, and empirically using collected data. Our evaluation suggests that the new model can be reliably used for model fitting across a variety of conditions. Our work provides a foundation for modeling measurements in flow cytometry experiments thus facilitating progress in quantitative studies of lymphocyte responses.

Project description:BackgroundStochastic effects can be important for the behavior of processes involving small population numbers, so the study of stochastic models has become an important topic in the burgeoning field of computational systems biology. However analysis techniques for stochastic models have tended to lag behind their deterministic cousins due to the heavier computational demands of the statistical approaches for fitting the models to experimental data. There is a continuing need for more effective and efficient algorithms. In this article we focus on the parameter inference problem for stochastic kinetic models of biochemical reactions given discrete time-course observations of either some or all of the molecular species.ResultsWe propose an algorithm for inference of kinetic rate parameters based upon maximum likelihood using stochastic gradient descent (SGD). We derive a general formula for the gradient of the likelihood function given discrete time-course observations. The formula applies to any explicit functional form of the kinetic rate laws such as mass-action, Michaelis-Menten, etc. Our algorithm estimates the gradient of the likelihood function by reversible jump Markov chain Monte Carlo sampling (RJMCMC), and then gradient descent method is employed to obtain the maximum likelihood estimation of parameter values. Furthermore, we utilize flux balance analysis and show how to automatically construct reversible jump samplers for arbitrary biochemical reaction models. We provide RJMCMC sampling algorithms for both fully observed and partially observed time-course observation data. Our methods are illustrated with two examples: a birth-death model and an auto-regulatory gene network. We find good agreement of the inferred parameters with the actual parameters in both models.ConclusionsThe SGD method proposed in the paper presents a general framework of inferring parameters for stochastic kinetic models. The method is computationally efficient and is effective for both partially and fully observed systems. Automatic construction of reversible jump samplers and general formulation of the likelihood gradient function makes our method applicable to a wide range of stochastic models. Furthermore our derivations can be useful for other purposes such as using the gradient information for parametric sensitivity analysis or using the reversible jump samplers for full Bayesian inference. The software implementing the algorithms is publicly available at http://cbcl.ics.uci.edu/sgd.

Project description:BackgroundUpon antigen encounter, naïve B lymphocytes differentiate into antibody-secreting plasma cells. This humoral immune response is suppressed by the environmental contaminant 2,3,7,8-tetrachlorodibenzo-p-dioxin (TCDD) and other dioxin-like compounds, which belong to the family of aryl hydrocarbon receptor (AhR) agonists.ResultsTo achieve a better understanding of the immunotoxicity of AhR agonists and their associated health risks, we have used computer simulations to study the behavior of the gene regulatory network underlying B cell terminal differentiation. The core of this network consists of two coupled double-negative feedback loops involving transcriptional repressors Bcl-6, Blimp-1, and Pax5. Bifurcation analysis indicates that the feedback network can constitute a bistable system with two mutually exclusive transcriptional profiles corresponding to naïve B cells and plasma cells. Although individual B cells switch to the plasma cell state in an all-or-none fashion when stimulated by the polyclonal activator lipopolysaccharide (LPS), stochastic fluctuations in gene expression make the switching event probabilistic, leading to heterogeneous differentiation response among individual B cells. Moreover, stochastic gene expression renders the dose-response behavior of a population of B cells substantially graded, a result that is consistent with experimental observations. The steepness of the dose response curve for the number of plasma cells formed vs. LPS dose, as evaluated by the apparent Hill coefficient, is found to be inversely correlated to the noise level in Blimp-1 gene expression. Simulations illustrate how, through AhR-mediated repression of the AP-1 protein, TCDD reduces the probability of LPS-stimulated B cell differentiation. Interestingly, stochastic simulations predict that TCDD may destabilize the plasma cell state, possibly leading to a reversal to the B cell phenotype.ConclusionOur results suggest that stochasticity in gene expression, which renders a graded response at the cell population level, may have been exploited by the immune system to launch humoral immune response of a magnitude appropriately tuned to the antigen dose. In addition to suppressing the initiation of the humoral immune response, dioxin-like compounds may also disrupt the maintenance of the acquired immunity.

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Stochastic models of lymphocyte proliferation and death.

Publications

Stochastic models of lymphocyte proliferation and death.

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