Mechanism of cell-mediated cytotoxicity at the single cell level. VIII. Kinetics of lysis of target cells bound by more than one cytotoxic T lymphocyte

We measured the effects of having multiple cytotoxic T lymphocytes (CTL) bound to one target cell by using the single-cell cytotoxicity in agarose assay. We found that even though there is variability in the time at which individual target cells are lysed, we can identify a general trend: the mean rate of lysis increases with the number of CTL bound per target cell, reaching a maximum when the CTL-target cell ratio is three. Combining a quantitative model for the rate of lethal hitting in multicellular conjugates with a multi-event model for the rate of target cell disintegration, we developed a new multistage kinetic model for predicting the rate of target cell lysis in multiple lymphocyte-target cell conjugates. The variability in the time at which target cells are hit and the variability in the time until they disintegrate are incorporated into the model. By analyzing our measured data in the context of the multistage kinetic model, we were able to estimate via nonlinear least squares regression the target cell disintegration rate, but not the lethal hitting rate. Lethal hitting appeared to be too fast, when compared with disintegration, to significantly affect the time of target cell lysis. By using previously determined values of the lethal hitting rate for single lymphocyte-target cell conjugates and by postulating that lymphocytes act independently of each other in delivering lethal hits, we were able to estimate the rate at which target cells are hit in multiple-lymphocyte single target cell conjugates. By using this estimate of the lethal hitting rate and the regression estimate of the disintegration rate, the multistage kinetic model gave a quantitative fit to our data. From this analysis, we found that the rate at which a target cell disintegrates after being lethally hit increases with the number of CTL per conjugate. This result is quite surprising, because once the first hit has been received, a target cell can disintegrate in a killer cell-independent manner. Under the conditions of our experiment, it appears as if target cell disintegration is not killer cell-independent. Furthermore, our analysis of the time course of target cell disintegration suggests that the process is not governed by simple first order kinetics, but rather by a more complex multistep mechanism.     

Kinetics of cell-mediated cytotoxicity: Stochastic and deterministic multistage models

One of the body's major defenses against viral diseases and tumors is the killing of abnormal cells by host defense cells, such as T lymphocytes. The mechanism by which killing is accomplished is unknown. Here we develop both stochastic and deterministic models for the kinetics of killing in aggregates which contain a single lymphocyte and multiple target cells (LT_n conjugates), as might be seen early in an immune response, and in aggregates containing multiple lymphocytes and a single target cell (L_nT conjugates), which is characteristic of the late phase of a successful immune response. Comparing our models with data, we rule out the possibility of certain classes of lytic mechanisms and draw attention to the characteristics of likely mechanisms. Our stochastic model can be viewed as a specialized application of queueing theory to cell biology. For certain choices of arrival-time and service-time distributions, we find an exact correspondence between our stochastic and deterministic models.     

A moving boundary model of acrosomal elongation

A sperm penetrates an egg by extending a long, actin-filled tube known as the acrosomal process. This simple example of biomotility is one of the most dramatic. In Thyone, a 90 m process can extend in less than 10 s. Experiments have shown that actin monomers stored in the base of the sperm are transported to the growing tip of the acrosomal process where they add to the ends of the existing filaments.The force that drives the elongation of the acrosomal process has not yet been identified although the most frequently discussed candidate is the actin polymerization reaction. Developing what we believe are realistic moving boundary models of diffusion limited actin fiber polymerization, we show that actin filament growth occurs too slowly to drive acrosomal elongation. We thus believe that other forces, such as osmotically driven water flow, must play an important role in causing the elongation. We conjecture that actin polymerization merely follows to give the appropriate shape to the growing structure and to stabilize the structure once water flow ceases.Work partially supported by the United States Department of Energy     

Immune network behavior—II. From oscillations to chaos and stationary states

Two types of behavior have been previously reported in models of immune networks. The typical behavior of simple models, which involve B cells only, is stationary behavior involving several steady states. Finite amplitude perturbations may cause the model to switch between different equilibria. The typical behavior of more realistic models, which involve both B cells and antibody, consists of autonomous oscillations and/or chaos. While stationary behavior leads to easy interpretations in terms of idiotypic memory, oscillatory behavior seems to be in better agreement with experimental data obtained in unimmunized animals. Here we study a series of models of the idiotypic interaction between two B cell clones. The models differ with respect to the incorporation of antibodies, B cell maturation and compartmentalization. The most complicated model in the series has two realistic parameter regimes in which the behavior is respectively stationary and chaotic. The stability of the equilibrium states and the structure and interactions of the stable and unstable manifolds of the saddle-type equilibria turn out to be factors influencing the model's behavior. Whether or not the model is able to attain any form of sustained oscillatory behavior, i.e. limit cycles or chaos, seems to be determined by (global) bifurcations involving the stable and unstable manifolds of the equilibrium states. We attempt to determine whether such behavior should be expected to be attained from reasonable initial conditions by incorporating an immune response to an antigen in the model. A comparison of the behavior of the model with experimental data from the literature provides suggestions for the parameter regime in which the immune system is operating.     

Plasmid copy number control: a case study of the quasi-steady-state assumption.

Perelson & Brendel (1989, J. molec. Biol. 208, 245-255) have proposed kinetic models for the control of plasmid copy number, based on experiments by J. Tomizawa and his associates. The quasi-steady-state assumptions (QSSA) made in the analysis of these models are justified in the present paper, thereby providing an example of how QSSA can provide a powerful and reliable tool in the analysis of biological kinetics.     

Modeling immune reactivity in secondary lymphoid organs

Models of the dynamical interactions important in generating immune reactivity have generally assumed that the immune system is a single well-stirred compartment. Here we explicitly take into account the compartmentalized nature of the immune system and show that qualitative conclusions, such as the stability of the immune steady state, depend on architectural details. We examine a simple model idiotypic network involving only two types of B cells and antibody molecules. We show, for model parameters used by De Boeret al. (1990,Chem. Eng. Sci. 45, 2375–2382), that the immune steady state is unstable in a one compartmental model but stable in a two compartment model that contains both a lymphoid organ, such as the spleen, and the circulatory system. This work was performed under the auspices of the U.S. Department of Energy.     

Mathematical analysis of delay differential equation models of HIV-1 infection

Models of HIV-1 infection that include intracellular delays are more accurate representations of the biology and change the estimated values of kinetic parameters when compared to models without delays. We develop and analyze a set of models that include intracellular delays, combination antiretroviral therapy, and the dynamics of both infected and uninfected T cells. We show that when the drug efficacy is less than perfect the estimated value of the loss rate of productively infected T cells, delta, is increased when data is fit with delay models compared to the values estimated with a non-delay model. We provide a mathematical justification for this increased value of delta. We also provide some general results on the stability of non-linear delay differential equation infection models.     

Pattern formation in one- and two-dimensional shape-space models of the immune system.

A large-scale model of the immune network is analyzed, using the shape-space formalism. In this formalism, it is assumed that the immunoglobulin receptors on B cells can be characterized by their unique portions, or idiotypes, that have shapes that can be represented in a space of a small finite dimension. Two receptors are assumed to interact to the extent that the shapes of their idiotypes are complementary. This is modeled by assuming that shapes interact maximally whenever their coordinates in the space-space are equal and opposite, and that the strength of interaction falls off for less complementary shapes in a manner described by a Gaussian function of the Euclidean "distance" between the pair of interacting shapes. The degree of stimulation of a cell when confronted with complementary idiotypes is modeled using a log bell-shaped interaction function. This leads to three possible equilibrium states for each clone: a virgin, an immune, and a suppressed state. The stability properties of the three possible homogeneous steady states of the network are examined. For the parameters chosen, the homogeneous virgin state is stable to both uniform and sinusoidal perturbations of small amplitude. A sufficiently large perturbation will, however, destabilize the virgin state and lead to an immune reaction. Thus, the virgin system is both stable and responsive to perturbations. The homogeneous immune state is unstable to both uniform and sinusoidal perturbations, whereas the homogeneous suppressed state is stable to uniform, but unstable to sinusoidal, perturbations. The non-uniform patterns that arise from perturbations of the homogeneous states are examined numerically. These patterns represent the actual immune repertoire of an animal, according to the present model. The effect of varying the standard deviation sigma of the Gaussian is numerically analyzed in a one-dimensional model. If sigma is large compared to the size of the shape-space, the system attains a fixed non-uniform equilibrium. Conversely if sigma is small, the system attains one out of many possible non-uniform equilibria, with the final pattern depending on the initial conditions. This demonstrates the plasticity of the immune repertoire in this shape-space model. We describe how the repertoire organizes itself into large clusters of clones having similar behavior. These results are extended by analyzing pattern formation in a two-dimensional (2-D) shape-space.(ABSTRACT TRUNCATED AT 400 WORDS)