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Specific lysis modeling and parameter extraction from experimental data

Dmitry Shchelokov
September 28, 2021

Introduction

Specific lysis is a complex multi-step process of target cell death mediated by cytotoxic granules containing enzymes (perforins, granzymes) released by effector cells after engagement with a target. Indeed, the probability of target lysis should depend on the number of surrounding effector cells (i.e., effector to target ratio). Since the cell-cell interactions during target killing are discrete processes and an exact number of cell aggregates is unknown, a detailed description of cell lysis is complicated and contains plenty of parameters. This work aims to provide simple biologically relevant equations describing cell lysis dependence on both time and effector-target ratio.

Derivation

Specific lysis may be considered as first-order process of target cell death after engagement with one or more effector cells due to release of cytotoxic proteins:

where [〖T:E〗_i ] is number of target cell aggregates with i effector cells and 〖k_kill〗_i is rate constant of specific lysis which also should be dependent on number of effector cells surrounding target. Since the cell-cell interactions during target killing are a discrete processes and exact number and size of certain cell aggregates is unknown, detailed description of cell aggregates formation based on the law of mass action is unable to provide simple model avoiding rough assumptions that may affect the simulation results. Thereby, an alternative approach is required to describe this process in simplified manner with minimal number of unknown parameters.

Clearly, that probability of target cell to be lysed depends on the number of effector cells in its surrounding. If we consider the well-mixed system consisting of target and effector cells (purified-effector case), at any given time target cell is surrounded by number of cells of two types. Hence, the probability that any target cell has the certain set of neighbors may be expressed in the following way [PMID: 21440033]:

where i is the number of effector cells surrounding target cell; n is the maximal number of the possible cell-to-cell contacts (neighbors in this consideration); E and T is number of effector and target cells, respectively. Assuming that i effectors kill i-times faster than one effector cell we may write new expression for the specific lysis rate and make some algebraic rearrangements to obtain more compact form:

where the last term in final equation is referring to the influence of the effector-target ratio on the specific lysis which is usually observed in publications on in vitro experiments with cell- mediated cytotoxicity.

Specific lysis is a complex multi-step process of target cell death mediated by cytotoxic granules containing enzymes (perforins, granzymes, etc.) that released by effector cells after engagement with target. Clearly, that rate of target cell lysis should depend on activity of these enzymes:

where is a maximal rate constant of specific lysis and is an activity of enzymes which gave half-maximum rate of killing.

On the other hand, the activity of released enzymes relative to certain target cells should be proportional to the number of surrounding effector cells which actually corresponds to the effector-target ratio:

Hence, we got the explicit dependence on effector-target ratio and may explore some boundary behaviors of specific lysis rate as tends to zero or to infinity:

Thus, the derived equation demonstrates saturation behavior at increasing concentration of both effector and target cells in accordance with experimental observations [PMID: 408421].

Equations for description of in vitro experiments

*EXPERIMENTAL DESIGN*

Assumptions based on experimental conditions (incubation time ~4-6 hours):

  • effector and target cells do not proliferate during short-term experiments
  • target cells may undergo spontaneous lysis
  • ratio does not change during time course of experiments, i.e. target engagement/killing induces release of cytotoxic proteins and it takes a time for effectors to restore their cytotoxic activity [PMID: 25305322 – Fig. 4A]

ODE for target cells dynamics in these experiments:

Analytical solution:

Specific lysis, by definition, is calculated on the basis of number of dead cells or concentration of substances released during cell death (dye, isotope, etc.) [PMID: 408421]:

where [test] dead cells in experimental sample, [sp] dead cells in sample without effectors, [max] maximal number of dead cells when all of them are lysed. Hence, substituting analytical solutions for target cells:

Parameters extraction from experimental data

Hence, it is possible to estimate unknown parameters

1. If we have data in range of up to:

2. If we have data in the certain range of values:

Let’s choose two values (1) and (2) and express the ratio of corresponding :

From this equation the may be obtained:

Knowing value, may be calculated at any other value:

Model verification

Example 1

Let’s analyze data from the study [PMID: 27454297– Fig. 5 B]. Authors assessed the cytolysis of target cells after 18 hours of incubation with CAR-T cells. Here, we have range of ratios from 1 to high values. Applying formulas described above:

Example 2

Let’s analyze data from the study [PMID: 408421 – Fig. 2 A-B]. Authors assessed the number of dead target cells depending on number of both effector and target cells after 4.5 hours of incubation. The number of dead cells may be expressed as following:

Also, explicit dependence on time was plotted against experimental data [PMID: 408421 – Fig. 1A]. For this data set was re-estimated, assuming similar value:

Example 3

Let’s analyze data from the study [PMID: 29499048 – Fig. 2 C]. Authors assessed the kinetics of cytotoxicity depending on E:T ratio during incubation of target cells with NK cells. Here, we have the range of e/t ratios from 0.625 to 10 at two time points.

Moreover, authors assessed the time at which 50% lysis of target cells occurs, i.e. 50% killing time, depending on E:T ratio [PMID: 29499048 – Fig. 2 D].

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