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Cytotoxic T lymphocyte (CTL)-mediated killing involves the formation of a synapse with a target cell, followed by delivery of perforin and granzymes. Previously, we derived a general functional response for CTL killing while considering that CTLs form stable synapses (i.e., single-stage) and that the number of conjugates remains at steady state. However, the killing of target cells sometimes requires multiple engagements (i.e., multistage). To study how multistage killing and a lack of steady state influence the functional response, we here analyze a set of differential equations as well as simulations employing the cellular Potts model, in both cases describing CTLs that kill target cells. We find that at steady state the total killing rate (i.e., the number of target cells killed by all CTLs) is well described by the previously derived double saturation function. Compared to single-stage killing, the total killing rate during multistage killing saturates at higher CTL and target cell densities. Importantly, when the killing is measured before the steady state is approached, a qualitatively different functional response emerges for two reasons: First, the killing signal of each CTL gets diluted over several targets and because this dilution effect is strongest at high target cell densities; this can result in a peak in the dependence of the total killing rate on the target cell density. Second, the total killing rate exhibits a sigmoid dependence on the CTL density when killing is a multistage process, because it takes typically more than one CTL to kill a target. In conclusion, a sigmoid dependence of the killing rate on the CTLs during initial phases of killing may be indicative of a multistage killing process. Observation of a sigmoid functional response may thus arise from a dilution effect and is not necessarily due to cooperative behavior of the CTLs.

Cytotoxic T lymphocytes (CTLs) kill virus-infected cells and tumor cells, and play a critical role in immune protection. Our knowledge of how the CTL killing efficiency varies with CTL and target cell numbers is limited. Here, we simulate a region of lymphoid tissue using a cellular Potts model to characterize the functional response of CTL killing of target cells, and find that the total killing rate saturates both with the CTL and the target cell densities. The relative saturation in CTL and target cell densities is determined by whether a CTL can kill multiple target cells at the same time, and whether a target cell can be killed by many CTLs together. We find that all the studied regimes can be well described by a double-saturation (DS) function with two different saturation constants. We show that this DS model can be mechanistically derived for the cases where target cells are killed by a single CTL. For the other cases, a biological interpretation of the parameters is still possible. Our results imply that this DS function can be used as a tool to predict the cellular interactions in cytotoxicity data.

A wave of apoptosis (programmed cell death) occurs normally in pancreatic β-cells of newborn mice. We previously showed that macrophages from non-obese diabetic (NOD) mice become activated more slowly and engulf apoptotic cells at a lower rate than macrophages from control (Balb/c) mice. It has been hypothesized that this low clearance could result in secondary necrosis, escalating inflammation and self-antigen presentation that later triggers autoimmune, Type 1 diabetes (T1D). We here investigate whether this hypothesis could offer a reasonable and parsimonious explanation for onset of T1D in NOD mice. We quantify variants of the Copenhagen model(Freiesleben De Blasio et al. 1999 Diabetes48, 1677), based on parameters from NOD and Balb/c experimental data. We show that the original Copenhagen model fails to explain observed phenomena within a reasonable range of parameter values, predicting an unrealistic all-or-none disease occurrence for both strains. However, if we take into account that, in general, activated macrophages produce harmful cytokines onlywhen engulfing necrotic (but not apoptotic) cells, then the revised model becomes qualitatively and quantitatively reasonable. Further, we show that known differences between NOD and Balb/c mouse macrophage kinetics are large enough to account for the fact that an apoptotic wave can trigger escalating inflammatory response in NOD, but not Balb/c mice. In Balb/c mice, macrophages clear the apoptotic wave so efficiently, that chronic inflammation is prevented.

Dictyostelium discoideumslugs show a pronounced thermotaxis. We have modelled the motion of the D. discoideumslug in the absence and in the presence of a thermal gradient. Our model is an extension of the hybrid cellular automata/partial differential equation model, as formulated by Savill and Hogeweg [J. theor. Biol., (1997) 184, 229–235]. The modelled slugs maintain their shape and crawl, with a velocity depending on slug size, as found in experiments. Moreover, they show thermotactic behaviour: independent of the initial orientation, after some transient process, the slugs start moving along the temperature gradient. The slug behaviour in our model is due to the collective behaviour of the amoebae. Individual amoebae can neither respond to a shallow temperature gradient, nor show differentiation in motion velocity. The behaviour is achieved by a modification of the cyclic AMP waves: differences in temperature alter the excitability of the cell, and thereby the shape of the cyclic AMP wave. Chemotaxis towards cyclic AMP causes the slug to turn. We show that the mechanism still functions at very low signal-to-noise ratios.

We examined the responses to transparent motion of complex cells in cat area 17 which show directional selectivity to moving random pixel arrays (RPAs). The response to an RPA moving in the cell's preferred direction is inhibited when a second RPA is transparently moving in another direction. The inhibition by the second pattern is quantified as a function of its direction. The response to a pattern moving in the preferred direction is never completely suppressed, not even when a second pattern is moving transparently in the opposite direction.