A mean-field description for the expansion kinetics of activated T cell populations.

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.

Virtual Populations for Quantitative Systems Pharmacology Models.

Quantitative systems pharmacology (QSP) places an emphasis on dynamic systems modeling, incorporating considerations from systems biology modeling and pharmacodynamics. The goal of QSP is often to quantitatively predict the effects of clinical therapeutics, their combinations, and their doses on clinical biomarkers and endpoints. In order to achieve this goal, strategies for incorporating clinical data into model calibration are critical. Virtual population (VPop) approaches facilitate model calibration while faced with challenges encountered in QSP model application, including modeling a breadth of clinical therapies, biomarkers, endpoints, utilizing data of varying structure and source, capturing observed clinical variability, and simulating with models that may require more substantial computational time and resources than often found in pharmacometrics applications. VPops are frequently developed in a process that may involve parameterization of isolated pathway models, integration into a larger QSP model, incorporation of clinical data, calibration, and quantitative validation that the model with the accompanying, calibrated VPop is suitable to address the intended question or help with the intended decision. Here, we introduce previous strategies for developing VPops in the context of a variety of therapeutic and safety areas: metabolic disorders, drug-induced liver injury, autoimmune diseases, and cancer. We introduce methodological considerations, prior work for sensitivity analysis and VPop algorithm design, and potential areas for future advancement. Finally, we give a more detailed application example of a VPop calibration algorithm that illustrates recent progress and many of the methodological considerations. In conclusion, although methodologies have varied, VPop strategies have been successfully applied to give valid clinical insights and predictions with the assistance of carefully defined and designed calibration and validation strategies. While a uniform VPop approach for all potential QSP applications may be challenging given the heterogeneity in use considerations, we anticipate continued innovation will help to drive VPop application for more challenging cases of greater scale while developing new rigorous methodologies and metrics.

PIKES Analysis Reveals Response to Degraders and Key Regulatory Mechanisms of the CRL4 Network.

Co-opting Cullin4 RING ubiquitin ligases (CRL4s) to inducibly degrade pathogenic proteins is emerging as a promising therapeutic strategy. Despite intense efforts to rationally design degrader molecules that co-opt CRL4s, much about the organization and regulation of these ligases remains elusive. Here, we establish protein interaction kinetics and estimation of stoichiometries (PIKES) analysis, a systematic proteomic profiling platform that integrates cellular engineering, affinity purification, chemical stabilization, and quantitative mass spectrometry to investigate the dynamics of interchangeable multiprotein complexes. Using PIKES, we show that ligase assemblies of Cullin4 with individual substrate receptors differ in abundance by up to 200-fold and that Cand1/2 act as substrate receptor exchange factors. Furthermore, degrader molecules can induce the assembly of their cognate CRL4, and higher expression of the associated substrate receptor enhances degrader potency. Beyond the CRL4 network, we show how PIKES can reveal systems level biochemistry for cellular protein networks important to drug development.

Cand1-Mediated Adaptive Exchange Mechanism Enables Variation in F-Box Protein Expression.

Skp1⋅Cul1⋅F-box (SCF) ubiquitin ligase assembly is regulated by the interplay of substrate binding, reversible Nedd8 conjugation on Cul1, and the F-box protein (FBP) exchange factors Cand1 and Cand2. Detailed investigations into SCF assembly and function in reconstituted systems and Cand1/2 knockout cells informed the development of a mathematical model for how dynamical assembly of SCF complexes is controlled and how this cycle is coupled to degradation of an SCF substrate. Simulations predicted an unanticipated hypersensitivity of Cand1/2-deficient cells to FBP expression levels, which was experimentally validated. Together, these and prior observations lead us to propose the adaptive exchange hypothesis, which posits that regulation of the koff of an FBP from SCF by the actions of substrate, Nedd8, and Cand1 molds the cellular repertoire of SCF complexes and that the plasticity afforded by this exchange mechanism may enable large variations in FBP expression during development and in FBP gene number during evolution.

Analysis of network motifs in cellular regulation: Structural similarities, input-output relations and signal integration.

Much of the complexity of regulatory networks derives from the necessity to integrate multiple signals and to avoid malfunction due to cross-talk or harmful perturbations. Hence, one may expect that the input-output behavior of larger networks is not necessarily more complex than that of smaller network motifs which suggests that both can, under certain conditions, be described by similar equations. In this review, we illustrate this approach by discussing the similarities that exist in the steady state descriptions of a simple bimolecular reaction, covalent modification cycles and bacterial two-component systems. Interestingly, in all three systems fundamental input-output characteristics such as thresholds, ultrasensitivity or concentration robustness are described by structurally similar equations. Depending on the system the meaning of the parameters can differ ranging from protein concentrations and affinity constants to complex parameter combinations which allows for a quantitative understanding of signal integration in these systems. We argue that this approach may also be extended to larger regulatory networks.

Trade-off and flexibility in the dynamic regulation of the cullin-RING ubiquitin ligase repertoire.

Cullin-RING ubiquitin ligases (CRLs) catalyze the ubiquitylation of substrates many of which are degraded by the 26S proteasome. Their modular architecture enables recognition of numerous substrates via exchangeable substrate receptors that competitively bind to a cullin scaffold with high affinity. Due to the plasticity of these interactions there is ongoing uncertainty how cells maintain a flexible CRL repertoire in view of changing substrate loads. Based on a series of in vivo and in vitro studies, different groups proposed that the exchange of substrate receptors is mediated by a protein exchange factor named Cand1. Here, we have performed mathematical modeling to provide a quantitative underpinning of this hypothesis. First we show that the exchange activity of Cand1 necessarily leads to a trade-off between high ligase activity and fast receptor exchange. Supported by measurements we argue that this trade-off yields an optimal Cand1 concentration in cells where the time scale for substrate degradation becomes minimal. In a second step we show through simulations that (i) substrates bias the CRL repertoire leading to preferential assembly of ligases for which substrates are available and (ii) differences in binding affinities or substrate receptor abundances create a temporal hierarchy for the degradation of substrates. Finally, we compare the Cand1-mediated exchange cycle with an alternative architecture lacking Cand1 which indicates superiority of a system with exchange factor if substrate receptors bind substrates and the cullin scaffold in a random order. Together, our results provide general constraints for the operating regimes of molecular exchange systems and suggest that Cand1 endows the CRL network with the properties of an "on demand" system allowing cells to dynamically adjust their CRL repertoire to fluctuating substrate abundances.

Operating regimes of covalent modification cycles at high enzyme concentrations.

The Goldbeter-Koshland model has been a paradigm for ultrasensitivity in biological networks for more than 30 years. Despite its simplicity the validity of this model is restricted to conditions when the substrate is in excess over the converter enzymes - a condition that is easy to satisfy in vitro, but which is rarely satisfied in vivo. Here, we analyze the Goldbeter-Koshland model by means of the total quasi-steady state approximation which yields a comprehensive classification of the steady state operating regimes under conditions when the enzyme concentrations are comparable to or larger than that of the substrate. Where possible we derive simple expressions characterizing the input-output behavior of the system. Our analysis suggests that enhanced sensitivity occurs if the concentration of at least one of the converter enzymes is smaller (but not necessarily much smaller) than that of the substrate and if that enzyme is saturated. Conversely, if both enzymes are saturated and at least one of the enzyme concentrations exceeds that of the substrate the system exhibits concentration robustness with respect to changes in that enzyme concentration. Also, depending on the enzyme's saturation degrees and the ratio between their maximal reaction rates the total fraction of phosphorylated substrate may increase, decrease or change nonmonotonically as a function of the total substrate concentration. The latter finding may aid the interpretation of experiments involving genetic perturbations of enzyme and substrate abundances.

Bistability and Nonmonotonic Induction of the lac Operon in the Natural Lactose Uptake System.

The Escherichia coli lac operon is regulated by a positive feedback loop whose potential to generate an all-or-none response in single cells has been a paradigm for bistable gene expression. However, so far bistable lac induction has only been observed using gratuitous inducers, raising the question about the biological relevance of bistable lac induction in the natural setting with lactose as the inducer. In fact, the existing experimental evidence points to a graded rather than an all-or-none response in the natural lactose uptake system. In contrast, predictions based on computational models of the lactose uptake pathway remain controversial. Although some argue in favor of bistability, others argue against it. Here, we reinvestigate lac operon expression in single cells using a combined experimental/modeling approach. To this end, we parameterize a well-supported mathematical model using transient measurements of LacZ activity upon induction with different amounts of lactose. The resulting model predicts a monostable induction curve for the wild-type system, but indicates that overexpression of the LacI repressor would drive the system into the bistable regime. Both predictions were confirmed experimentally supporting the view that the wild-type lac induction circuit generates a graded response rather than bistability. More interestingly, we find that the lac induction curve exhibits a pronounced maximum at intermediate lactose concentrations. Supported by our data, a model-based analysis suggests that the nonmonotonic response results from saturation of the LacI repressor at low inducer concentrations and dilution of Lac enzymes due to an increased growth rate beyond the saturation point. We speculate that the observed maximum in the lac expression level helps to save cellular resources by limiting Lac enzyme expression at high inducer concentrations.

Analysis of substrate competition in regulatory network motifs: Stimulus-response curves, thresholds and ultrasensitivity.

In the simplest case, substrate competition arises if two ligands compete for access to a single binding site of a receptor protein (or enzyme). If the two ligands exhibit different binding affinities the competition becomes biased: as long as the receptor concentration remains lower than that of the high-affinity ligand the latter blocks all of the available binding sites so that the concentration of the complex comprising the low-affinity ligand remains low. The latter only rises if the receptor concentration is increased beyond that of the high-affinity ligand. Depending on the binding affinity of the low-affinity ligand this increase may then occur in an ultrasensitive manner. Similar behavior has been observed in a phosphorylation/dephosphorylation cycle involved in cell-cycle regulation. However, a steady state analysis shows that in this case the threshold concentration is modulated by the catalytic rate constants for phosphorylation and dephosphorylation of the high-affinity substrate. As a consequence, there exists a trade-off between the dynamic range of the system as measured by the maximal phosphorylation level of the substrate and the sensitivity of the system as measured by the position of the threshold. Using the ratio of the binding affinities as a small parameter we derive explicit expressions for the stimulus-response curves as a function of the receptor (or enzyme) concentration as well as conditions for the occurrence of ultrasensitivity. Interestingly, the network motifs investigated in this study are described by structurally similar steady state equations indicating that the analysis presented here may be extendable for analyzing substrate competition in more complex regulatory networks.

Reciprocal regulation as a source of ultrasensitivity in two-component systems with a bifunctional sensor kinase.

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.

Sensitivity and robustness in covalent modification cycles with a bifunctional converter enzyme.

Regulation by covalent modification is a common mechanism to transmit signals in biological systems. The modifying reactions are catalyzed either by two distinct converter enzymes or by a single bifunctional enzyme (which may employ either one or two catalytic sites for its opposing activities). The reason for this diversification is unclear, but contemporary theoretical models predict that systems with distinct converter enzymes can exhibit enhanced sensitivity to input signals whereas bifunctional enzymes with two catalytic sites are believed to generate robustness against variations in system's components. However, experiments indicate that bifunctional enzymes can also exhibit enhanced sensitivity due to the zero-order effect, raising the question whether both phenomena could be understood within a common mechanistic model. Here, I argue that this is, indeed, the case. Specifically, I show that bifunctional enzymes with two catalytic sites can exhibit both ultrasensitivity and concentration robustness, depending on the kinetic operating regime of the enzyme's opposing activities. The model predictions are discussed in the context of experimental observations of ultrasensitivity and concentration robustness in the uridylylation cycle of the PII protein, and in the phosphorylation cycle of the isocitrate dehydrogenase, respectively.

Reciprocal enzyme regulation as a source of bistability in covalent modification cycles.

Covalent modification cycles (CMCs) are the building blocks of many regulatory networks in biological systems. Under proper kinetic conditions such mono-cyclic enzyme systems can show a higher sensitivity to effectors than enzymes subject to direct allosteric regulation. Using methods from reaction network theory it has been argued that CMCs can potentially exhibit multiple steady states if the converter enzymes are regulated in a reciprocal manner, but the underlying mechanism as well as the kinetic requirements for the emergence of such a behavior remained unclear. Here, we reinvestigate CMCs with reciprocal regulation of the converter enzymes for two common regulatory mechanisms: allosteric regulation and covalent modification. To analyze the steady state behavior of the corresponding mass-action equations, we derive reduced models by means of a quasi-steady state approximation (QSSA). We also derive reduced models using the total QSSA which often better reproduces the transient dynamics of enzyme-catalyzed reaction systems. Through a steady state analysis of the reduced models we show that the occurrence of bistability can be associated with the presence of a double negative feedback loop. We also derive constraints for the model parameters which might help to evaluate the potential significance of the mechanisms described here for the generation of bistability in natural systems. In particular, our results support the view of a possible bistable response in the metabolic PFK1/F1,6BPase cycle as observed experimentally in rat liver extracts, and it suggests an alternative view on the origin of bistability in the Cdk1-Wee1-Cdc25 system.

An extended model for the repression of photosynthesis genes by the AppA/PpsR system in Rhodobacter sphaeroides.

Purple bacteria derive energy from aerobic respiration or photosynthesis depending on the availability of oxygen and light. Under aerobic conditions, photosynthesis genes are specifically repressed by the PpsR protein. In Rhodobacter sphaeroides, the repressive action of PpsR is antagonized by the blue-light and redox-sensitive flavoprotein AppA, which sequesters PpsR under anaerobic conditions into transcriptionally inactive complexes. However, under semi-aerobic conditions, blue-light excitation of AppA causes the AppA-PpsR complexes to dissociate, again leading to a repression of photosynthesis genes. We have recently developed a simple mathematical model suggesting that this phenotype arises from the formation of a maximum in the response curve of reduced PpsR at intermediate oxygen concentrations. However, this model focused mainly on the oxygen-dependent interactions whereas light regulation was only implemented in a simplified manner. In the present study, we incorporate a more detailed mechanism for the light-dependent interaction between AppA and PpsR, which now allows for a direct comparison with experiments. Specifically, we take into account that, upon blue-light excitation, AppA undergoes a conformational change, creating a long-lived signalling state causing the dissociation of the AppA-PpsR complexes. The predictions of the extended model are found to be in good agreement with experimental results on the light-dependent repression of photosynthesis genes under semi-aerobic conditions. We also identify the potential kinetic and stoichiometric constraints that the interplay between light and redox regulation imposes on the functionality of the AppA/PpsR system, especially with respect to a possible bistable response.

Comment on "load-induced modulation of signal transduction networks": reconciling ultrasensitivity with bifunctionality?

Jiang et al. (Research Article, 11 October 2011, DOI: 10.1126/scisignal.2002152) used a combined experimental and computational modeling approach to study the dynamic response behavior of covalent modification cycles in the presence of downstream targets ("loads"). Despite remarkable agreement between experiments and model predictions, there exists an apparent discrepancy in their approach because the utilized theoretical model does not reflect the bifunctional nature of the enzyme system used in experiments. Furthermore, a simple extension of the model to the case of bifunctional enzymes yields predictions that are partially at variance with the experimental results. It seems that an appropriate mechanistic model would have to reconcile two apparent contradictory concepts: ultrasensitivity and bifunctionality.

Modeling the light- and redox-dependent interaction of PpsR/AppA in Rhodobacter sphaeroides.

Facultative photosynthetic bacteria switch their energy generation mechanism from respiration to photosynthesis depending on oxygen tension and light. Part of this transition is mediated by the aerobic transcriptional repressor PpsR. In Rhodobacter sphaeroides, the repressive action of PpsR is antagonized by the redox- and blue-light-sensitive flavoprotein AppA which results in a unique phenotype: the repression of photosynthesis genes at intermediate oxygen levels and high light intensity, which is believed to reduce the risk of photooxidative stress. To analyze the underlying mechanism we developed a simple mathematical model based on the AppA-dependent reduction of a disulfide bond in PpsR and the light-sensitive complex formation between the reduced forms of AppA and PpsR. A steady-state analysis shows that high light repression can indeed occur at intermediate oxygen levels if PpsR is reduced on a faster timescale than AppA and if the electron transfer from AppA to PpsR is effectively irreversible. The model further predicts that if AppA copy numbers exceed those of PpsR by at least a factor of two, the transition from aerobic to anaerobic growth mode can occur via a bistable regime. We provide necessary conditions for the emergence of bistability and discuss possible experimental verifications.

Brownian diffusion of ion channels in different membrane patch geometries.

We asymptotically calculate the spatially averaged mean first passage time (MFPT) of a diffusing channel protein in a finite membrane patch containing a small absorbing anchor site. Different two-dimensional membrane geometries are considered including a circular, a square-shaped, a rectangular, and a cylindrical domain. The asymptotic expressions are found to be in excellent agreement with results from Monte Carlo simulations if the radius of the diffusing protein is sufficiently small. For a larger radius, a simple correction to the asymptotic expressions is proposed. We show that the average MFPT for a circle and a square-shaped domain of the same area are approximately equal as long as the anchor site is close to the center of the domain. We also discuss how the average MFPT depends on the aspect ratio of a rectangular and a cylindrical domain. Among such domains with a fixed area, a minimal MFPT is obtained for the square-shaped domain.

Diffusive coupling can discriminate between similar reaction mechanisms in an allosteric enzyme system.

BACKGROUND: 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. RESULTS: 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. CONCLUSIONS: 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.

Inward rotating spiral waves in glycolysis.

We report on the first observation of inward rotating spiral waves (antispirals) in a biochemical reaction-diffusion system. Experiments are performed with extracts from yeast cells in an open spatial reactor. By increasing the protein concentration of the extract we observe a transition from outward to inward propagating waves of glycolytic activity. Numerical simulations with an allosteric model for the phosphofructokinase can reproduce these inward propagating waves over a wide range of parameters if the octameric structure of yeast phosphofructokinase is taken into account.

Investigating the effects of molecular crowding on Ca2+ diffusion using a particle-based simulation model.

Calcium ions (Ca(2+)) are an important second messenger in eucaryotic cells. They are involved in numerous physiological processes which are triggered by calcium signals in the form of local release events, temporal oscillations, or reaction-diffusion waves. The diffusive spread of calcium signals in the cytosol is strongly affected by calcium-binding proteins (buffers). In addition, the cytosol contains a large number of inert molecules and molecular structures which make it a crowded environment. Here, we investigate the effects of such excluded volumes on calcium diffusion in the presence of different kinds of buffers. We find that the contributions in slowing down Ca(2+) diffusion coming from buffering and molecular crowding are not additive, i.e., the reduction in Ca(2+) diffusivity due to crowding and buffering together is not the sum of each single contribution. In the presence of Ca(2+) gradients and high affinity mobile buffers the effective diffusion coefficient of Ca(2+) can be reduced by up to 60% in highly crowded environments. This suggests that molecular crowding may significantly affect the shape of Ca(2+) microdomains and wave propagation in cell types with high excluded volume fractions.

Spatial desynchronization of glycolytic waves as revealed by Karhunen-Loeve analysis.

The dynamics of glycolytic waves in a yeast extract have been investigated in an open spatial reactor. At low protein contents in the extract, we find a transition from inwardly moving target patterns at the beginning of the experiment to outwardly moving spiral- or circular-shaped waves at later stages. These two phases are separated by a transition phase of more complex spatiotemporal dynamics. We have analyzed the pattern dynamics in these three intervals at different spatial scales by means of a Karhunen-Loeve (KL) decomposition. During the initial phase of the experiment, the observed patterns are sufficiently described by the two dominant KL modes independently of the spatial scale. However, during the last stage of the experiment, at least 6 KL modes are needed to account for the observed patterns at spatial scales larger than 3 mm, while for smaller scales, 2 KL modes are still sufficient. This indicates that in the course of the experiment, the local glycolytic oscillators become desynchronized at spatial scales larger than 3 mm. Possible reasons for the desynchronization of the glycolytic waves are discussed.