The transition from simple periodic to bursting behavior in a three-dimensional model system of the hemin--hydrogen-peroxide--sulfite $p\mathrm{H}$ oscillator is investigated. A two-parameter continuation in the flow rate and the hemin decay rate is performed to identify the region of complex dynamics. The bursting oscillations emerge subsequent to a cascade of period-doubling bifurcations and the formation of a chaotic attractor in parameter space where they are found to be organized in periodic-chaotic progressions. This suggests that the bursting oscillations are not associated with phase-locked states on a two-torus. The bursting behavior is classified by a bifurcation analysis using the intrinsic slow-fast structure of the dynamics. In particular, we find a slowly varying quasispecies (i.e., a linear combination of two species) which acts as an ``internal'' or quasistatic bifurcation parameter for the remaining two-dimensional subsystem. A systematic two-parameter continuation in the internal parameter and one of the external bifurcation parameters reveals a transition in the bursting mechanism from sub-Hopf/fold-cycle to fold/sub-Hopf type. In addition, the slow-fast analysis provides an explanation for the origin of quasiperiodic behavior in the hemin system, even though the underlying mechanism might be of more general importance.
When lymphocytes encounter their cognate antigen, they become activated and undergo a limited number of cell divisions during which they differentiate into memory or effector cells or die. While the dynamics of individual cells are often heterogeneous, the expansion kinetics at the population level are highly reproducible, suggesting a mean-field description. To generate a finite division destiny, we consider two scenarios: Cells stop dividing after a certain number of iterations or their death rate increases with each cell division. The dynamics of the combined system can be mapped to a partial differential equation, and for a suitable choice of the activation rate, we obtain simple analytical solutions for the total cell number and the mean number of divisions per cell which can well describe the signal-dependent T cell expansion kinetics from in vitro experiments. Interestingly, only the division cessation mechanism yields an expression for the division destiny that does not contradict experiments. We show that the generation-dependent decrease of the division rate in individual cells leads to a time-dependent decrease at the population level which is consistent with a “time-to-die” control mechanism for the division destiny as suggested previously. We also derive mean-field equations for the total cell number which provide a basis for implementing T cell expansion kinetics into quantitative systems pharmacology models for immuno-oncology and CAR-T cell therapies.
A central question for the understanding of biological reaction networks is how a particular dynamic behavior, such as bistability or oscillations, is realized at the molecular level. So far this question has been mainly addressed in well-mixed reaction systems which are conveniently described by ordinary differential equations. However, much less is known about how molecular details of a reaction mechanism can affect the dynamics in diffusively coupled systems because the resulting partial differential equations are much more difficult to analyze. Motivated by recent experiments we compare two closely related mechanisms for the product activation of allosteric enzymes with respect to their ability to induce different types of reaction-diffusion waves and stationary Turing patterns. The analysis is facilitated by mapping each model to an associated complex Ginzburg-Landau equation. We show that a sequential activation mechanism, as implemented in the model of Monod, Wyman and Changeux (MWC), can generate inward rotating spiral waves which were recently observed as glycolytic activity waves in yeast extracts. In contrast, in the limiting case of a simple Hill activation, the formation of inward propagating waves is suppressed by a Turing instability. The occurrence of this unusual wave dynamics is not related to the magnitude of the enzyme cooperativity (as it is true for the occurrence of oscillations), but to the sensitivity with respect to changes of the activator concentration. Also, the MWC mechanism generates wave patterns that are more stable against long wave length perturbations. This analysis demonstrates that amplitude equations, which describe the spatio-temporal dynamics near an instability, represent a valuable tool to investigate the molecular effects of reaction mechanisms on pattern formation in spatially extended systems. Using this approach we have shown that the occurrence of inward rotating spiral waves in glycolysis can be explained in terms of an MWC, but not with a Hill mechanism for the activation of the allosteric enzyme phosphofructokinase. Our results also highlight the importance of enzyme oligomerization for a possible experimental generation of Turing patterns in biological systems.
Two-component signal transduction systems, where the phosphorylation state of a regulator protein is modulated by a sensor kinase, are common in bacteria and other microbes. In many of these systems, the sensor kinase is bifunctional catalyzing both, the phosphorylation and the dephosphorylation of the regulator protein in response to input signals. Previous studies have shown that systems with a bifunctional enzyme can adjust the phosphorylation level of the regulator protein independently of the total protein concentrations – a property known as concentration robustness. Here, I argue that two-component systems with a bifunctional enzyme may also exhibit ultrasensitivity if the input signal reciprocally affects multiple activities of the sensor kinase. To this end, I consider the case where an allosteric effector inhibits autophosphorylation and, concomitantly, activates the enzyme's phosphatase activity, as observed experimentally in the PhoQ/PhoP and NRII/NRI systems. A theoretical analysis reveals two operating regimes under steady state conditions depending on the effector affinity: If the affinity is low the system produces a graded response with respect to input signals and exhibits stimulus-dependent concentration robustness – consistent with previous experiments. In contrast, a high-affinity effector may generate ultrasensitivity by a similar mechanism as phosphorylation-dephosphorylation cycles with distinct converter enzymes. The occurrence of ultrasensitivity requires saturation of the sensor kinase's phosphatase activity, but is restricted to low effector concentrations, which suggests that this mode of operation might be employed for the detection and amplification of low abundant input signals. Interestingly, the same mechanism also applies to covalent modification cycles with a bifunctional converter enzyme, which suggests that reciprocal regulation, as a mechanism to generate ultrasensitivity, is not restricted to two-component systems, but may apply more generally to bifunctional enzyme systems.
Intracellular signaling gradients naturally arise through a local activation of a diffusible signaling molecule, e.g., by a localized kinase, and a subsequent deactivation at a distant cellular site, e.g., by a cytosolic phosphatase. Here, we consider a spherical cell containing a finite number of small spherical compartments where a signaling molecule becomes activated by a localized enzyme. For the activation rate two cases are considered: a saturated enzyme with a constant rate and an unsaturated enzyme with a linear rate. Using the method of matched asymptotic expansions, we derive approximate solutions of the steady-state diffusion equation with a linear deactivation rate to obtain the three-dimensional concentration profile of activated signaling molecules inside the cell. Depending on the diffusion length of the signaling molecule, the profiles decay either exponentially or algebraically, where the mode of decay is described by an associated Green's function. Our analysis provides simple expressions for the local concentration profile in the neighborhood of a signaling compartment, which can be used to estimate the amplitude and the extent of the gradients in the respective regimes. The global concentration profiles also depend on the cell size and the particular spatial arrangement of the compartments relative to each other and to the cell boundary.
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