Compact Modeling of Allosteric Multisite Proteins: Application to a Cell Size Checkpoint

We explore a framework to model the dose response of allosteric multisite phosphorylation proteins using a single auxiliary variable. This reduction can closely replicate the steady state behavior of detailed multisite systems such as the Monod-Wyman-Changeux allosteric model or rule-based models. Optimal ultrasensitivity is obtained when the activation of an allosteric protein by its individual sites is concerted and redundant. The reduction makes this framework useful for modeling and analyzing biochemical systems in practical applications, where several multisite proteins may interact simultaneously. As an application we analyze a newly discovered checkpoint signaling pathway in budding yeast, which has been proposed to measure cell growth by monitoring signals generated at sites of plasma membrane growth. We show that the known components of this pathway can form a robust hysteretic switch. In particular, this system incorporates a signal proportional to bud growth or size, a mechanism to read the signal, and an all-or-none response triggered only when the signal reaches a threshold indicating that sufficient growth has occurred.     

Multisite phosphorylation and network dynamics of cyclin-dependent kinase signaling in the eukaryotic cell cycle

Multisite phosphorylation of regulatory proteins has been proposed to underlie ultrasensitive responses required to generate nontrivial dynamics in complex biological signaling networks. We used a random search strategy to analyze the role of multisite phosphorylation of key proteins regulating cyclin-dependent kinase (CDK) activity in a model of the eukaryotic cell cycle. We show that multisite phosphorylation of either CDK, CDC25, wee1, or CDK-activating kinase is sufficient to generate dynamical behaviors including bistability and limit cycles. Moreover, combining multiple feedback loops based on multisite phosphorylation do not destabilize the cell cycle network by inducing complex behavior, but rather increase the overall robustness of the network. In this model we find that bistability is the major dynamical behavior of the CDK signaling network, and that negative feedback converts bistability into limit cycle behavior. We also compare the dynamical behavior of several simplified models of CDK regulation to the fully detailed model. In summary, our findings suggest that multisite phosphorylation of proteins is a critical biological mechanism in generating the essential dynamics and ensuring robust behavior of the cell cycle.     

Functional analysis of proteins and protein species using shotgun proteomics and linear mathematics

Covalent post-translational modification of proteins is the primary modulator of protein function in the cell. It greatly expands the functional potential of the proteome compared to the genome. In the past few years shotgun proteomics-based research, where the proteome is digested into peptides prior to mass spectrometric analysis has been prolific in this area. It has determined the kinetics of tens of thousands of sites of covalent modification on an equally large number of proteins under various biological conditions and uncovered a transiently active regulatory network that extends into diverse branches of cellular physiology. In this review, we discuss this work in light of the concept of protein speciation, which emphasizes the entire post-translationally modified molecule and its interactions and not just the modification site as the functional entity. Sometimes, particularly when considering complex multisite modification, all of the modified molecular species involved in the investigated condition, the protein species must be completely resolved for full understanding. We present a mathematical technique that delivers a good approximation for shotgun proteomics data.     

Emergence of bimodal cell population responses from the interplay between analog single-cell signaling and protein expression noise

ABSTRACT: BACKGROUND: Cell-to-cell variability in protein expression can be large, and its propagation through signaling networks affects biological outcomes. Here, we apply deterministic and probabilistic models and biochemical measurements to study how network topologies and cell-to-cell protein abundance variations interact to shape signaling responses. RESULTS: We observe bimodal distributions of extracellular signal-regulated kinase (ERK) responses to epidermal growth factor (EGF) stimulation, which are generally thought to indicate bistable or ultrasensitive signaling behavior in single cells. Surprisingly, we find that a simple MAPK/ERK-cascade model with negative feedback that displays graded, analog ERK responses at a single cell level can explain the experimentally observed bimodality at the cell population level. Model analysis suggests that a conversion of graded input--output responses in single cells to digital responses at the population level is caused by a broad distribution of ERK pathway activation thresholds brought about by cell-to-cell variability in protein expression. CONCLUSIONS: Our results show that bimodal signaling response distributions do not necessarily imply digital (ultrasensitive or bistable) single cell signaling, and the interplay between protein expression noise and network topologies can bring about digital population responses from analog single cell dose responses. Thus, cells can retain the benefits of robustness arising from negative feedback, while simultaneously generating population-level on/off responses that are thought to be critical for regulating cell fate decisions.     

Multiple sites of in vivo phosphorylation in the MDM2 oncoprotein cluster within two important functional domains

The MDM2 oncoprotein is a negative regulatory partner of the p53 tumour suppressor. MDM2 mediates ubiquitination of p53 and targets the protein to the cytoplasm for 26S proteosome-dependent degradation. In this paper, we show that MDM2 is modified in cultured cells by multisite phosphorylation. Deletion analysis of MDM2 indicated that the sites of modification fall into two clusters which map respectively within the N-terminal region encompassing the p53 binding domain and nuclear export sequence, and the central acidic domain that mediates p14(ARF) binding, p53 ubiquitination and cytoplasmic shuttling. The data are consistent with potential regulation of MDM2 function by multisite phosphorylation.     

Sensitivity analysis of metabolic cascades catalyzed by bifunctional enzymes

Covalent modification/demodification cycles are common in metabolism. When the modification and demodification steps are carried out by two independent enzymes, the degree of modification can be ultrasensitive to the total concentration of either catalyst. We recently showed that the degree of modification of a target molecule cannot exhibit ultrasensitivity to the free concentrations of effectors that decide whether a bifunctional enzyme acts as modifier or demodifier. However, here we can now demonstrate that the degree of modification of a target molecule can display ultrasensitivity to the total, rather than free, concentrations of such effectors. Our results clarify some general aspects of ultrasensitive responses to effectors, including competitive inhibitors, in mono-cyclic cascades.     

Substrate competition as a source of ultrasensitivity in the inactivation of Wee1

The mitotic regulators Wee1 and Cdk1 can inactivate each other through inhibitory phosphorylations. This double-negative feedback loop is part of a bistable trigger that makes the transition into mitosis abrupt and decisive. To generate a bistable response, some component of a double-negative feedback loop must exhibit an ultrasensitive response to its upstream regulator. Here, we experimentally demonstrate that Wee1 exhibits a highly ultrasensitive response to Cdk1. Several mechanisms can, in principle, give rise to ultrasensitivity, including zero-order effects, multisite phosphorylation, and competition mechanisms. We found that the ultrasensitivity in the inactivation of Wee1 arises mainly through two competition mechanisms: competition between two sets of phosphorylation sites in Wee1 and between Wee1 and other high-affinity Cdk1 targets. Based on these findings, we were able to reconstitute a highly ultrasensitive Wee1 response with purified components. Competition provides a simple way of generating the equivalent of a highly cooperative allosteric response.     

Ultrasensitivity in multisite phosphorylation of membrane-anchored proteins

Cellular signaling is initially confined to the plasma membrane, where the cytoplasmic tails of surface receptors and other membrane-anchored proteins are phosphorylated in response to ligand binding. These proteins often contain multiple phosphorylation sites that are regulated by membrane-confined enzymes. Phosphorylation of these proteins is thought to be tightly regulated, because they initiate and regulate signaling cascades leading to cellular activation, yet how their phosphorylation is regulated is poorly understood. Ultrasensitive or switchlike responses in their phosphorylation state are not expected because the modifying enzymes are in excess. Here, we describe a novel mechanism of ultrasensitivity exhibited by multisite membrane-anchored proteins, but not cytosolic proteins, even when enzymes are in excess. The mechanism underlying this concentration-independent ultrasensitivity is the local saturation of a single enzyme by multiple sites on the substrate. Local saturation is a passive process arising from slow membrane diffusion, steric hindrances, and multiple sites, and therefore may be widely applicable. Critical to this ultrasensitivity is the brief enzymatic inactivation that follows substrate modification. Computations are presented using ordinary differential equations and stochastic spatial simulations. We propose a new role, to our knowledge, for multisite membrane-anchored proteins, discuss experiments that can be used to probe the model, and relate our findings to previous theoretical work.     

Dose–response relationship from longitudinal data with response‐dependent dose modification using likelihood methods

In some clinical trials or clinical practice, the therapeutic agent is administered repeatedly, and doses are adjusted in each patient based on repeatedly measured continuous responses, to maintain the response levels in a target range. Because a lower dose tends to be selected for patients with a better outcome, simple summarizations may wrongly show a better outcome for the lower dose, producing an incorrect dose–response relationship. In this study, we consider the dose–response relationship under these situations. We show that maximum‐likelihood estimates are consistent without modeling the dose‐modification mechanisms when the selection of the dose as a time‐dependent covariate is based only on observed, but not on unobserved, responses, and measurements are generated based on administered doses. We confirmed this property by performing simulation studies under several dose‐modification mechanisms. We examined an autoregressive linear mixed effects model. The model represents profiles approaching each patient's asymptote when identical doses are repeatedly administered. The model takes into account the previous dose history and provides a dose–response relationship of the asymptote as a summary measure. We also examined a linear mixed effects model assuming all responses are measured at steady state. In the simulation studies, the estimates of both the models were unbiased under the dose modification based on observed responses, but biased under the dose modification based on unobserved responses. In conclusion, the maximum‐likelihood estimates of the dose–response relationship are consistent under the dose modification based only on observed responses.     

Multistep phosphorylation systems: tunable components of biological signaling circuits

Multisite phosphorylation of proteins is a powerful signal processing mechanism that plays crucial roles in cell division and differentiation as well as in disease. We recently demonstrated a novel phenomenon in cell cycle regulation by showing that cyclin-dependent kinase–dependent multisite phosphorylation of a crucial substrate is performed sequentially in the N-to-C terminal direction along the disordered protein. The process is controlled by key parameters, including the distance between phosphorylation sites, the distribution of serines and threonines in sites, and the position of docking motifs. According to our model, linear patterns of phosphorylation along disordered protein segments determine the signal-response function of a multisite phosphorylation switch. Here we discuss the general advantages and engineering principles of multisite phosphorylation networks as processors of kinase signals. We also address the idea of using the mechanistic logic of linear multisite phosphorylation networks to design circuits for synthetic biology applications.     

A Combination of Multisite Phosphorylation and Substrate Sequestration Produces Switchlike Responses

The phosphorylation of a protein on multiple sites has been proposed to promote the switchlike regulation of protein activity. Recent theoretical work, however, indicates that multisite phosphorylation, by itself, is less effective at creating switchlike responses than had been previously thought. The phosphorylation of a protein often alters its spatial localization, or its association with other proteins, and this sequestration can alter the accessibility of the substrate to the relevant kinases and phosphatases. Sequestration thus has the potential to interact with multisite phosphorylation to modulate ultrasensitivity and threshold. Here, using simple ordinary differential equations to represent phosphorylation, dephosphorylation, and binding/sequestration, we demonstrate that the combination of multisite phosphorylation and regulated substrate sequestration can produce a response that is both a good threshold and a good switch. Several strategies are explored, including both stronger and weaker sequestration with successive phosphorylations, as well as combinations that are more elaborate. In some strategies, such as when phosphorylation and dephosphorylation are segregated, a near-optimal switch is possible, where the effective Hill number equals the number of phosphorylation sites.     

The estimation of relative site variability among aligned homologous protein sequences

MOTIVATION: Maximum likelihood-based methods to estimate site by site substitution rate variability in aligned homologous protein sequences rely on the formulation of a phylogenetic tree and generally assume that the patterns of relative variability follow a pre-determined distribution. We present a phylogenetic tree-independent method to estimate the relative variability of individual sites within large datasets of homologous protein sequences. It is based upon two simple assumptions. Firstly that substitutions observed between two closely related sequences are likely, in general, to occur at the most variable sites. Secondly that non-conservative amino acid substitutions tend to occur at more variable sites. Our methodology makes no assumptions regarding the underlying pattern of relative variability between sites. RESULTS: We have compared, using data simulated under a non-gamma distributed model, the performance of this approach to that of a maximum likelihood method that assumes gamma distributed rates. At low mean rates of evolution our method inferred site by site relative substitution rates more accurately than the maximum likelihood approach in the absence of prior assumptions about the relationships between sequences. Our method does not directly account for the effects of mutational saturation, However, we have incorporated an 'ad-hoc' modification that allows the accurate estimation of relative site variability in fast evolving and saturated datasets.     

Molecular titration and ultrasensitivity in regulatory networks

Protein sequestration occurs when an active protein is sequestered by a repressor into an inactive complex. Using mathematical and computational modeling, we show how this regulatory mechanism (called “molecular titration”) can generate ultrasensitive or “all-or-none” responses that are equivalent to highly cooperative processes. The ultrasensitive nature of the input-output response is mainly determined by two parameters: the dimer dissociation constant and the repressor concentration. Because in vivo concentrations are tunable through a variety of mechanisms, molecular titration represents a flexible mechanism for generating ultrasensitivity. Using physiological parameters, we report how details of in vivo protein degradation affect the strength of the ultrasensitivity at steady state. Given that developmental systems often transduce signals into cell-fate decisions on timescales incompatible with steady state, we further examine whether molecular titration can produce ultrasensitive responses within physiologically relevant time intervals. Using Drosophila somatic sex determination as a developmental paradigm, we demonstrate that molecular titration can generate ultrasensitivity on timescales compatible with most cell-fate decisions. Gene duplication followed by loss-of-function mutations can create dominant negatives that titrate and compete with the original protein. Dominant negatives are abundant in gene regulatory circuits, and our results suggest that molecular titration might be generating an ultrasensitive response in these networks.     

Bistability from double phosphorylation in signal transduction. Kinetic and structural requirements

Previous studies have suggested that positive feedback loops and ultrasensitivity are prerequisites for bistability in covalent modification cascades. However, it was recently shown that bistability and hysteresis can also arise solely from multisite phosphorylation. Here we analytically demonstrate that double phosphorylation of a protein (or other covalent modification) generates bistability only if: (a) the two phosphorylation (or the two dephosphorylation) reactions are catalyzed by the same enzyme; (b) the kinetics operate at least partly in the zero-order region; and (c) the ratio of the catalytic constants of the phosphorylation and dephosphorylation steps in the first modification cycle is less than this ratio in the second cycle. We also show that multisite phosphorylation enlarges the region of kinetic parameter values in which bistability appears, but does not generate multistability. In addition, we conclude that a cascade of phosphorylation/dephosphorylation cycles generates multiple steady states in the absence of feedback or feedforward loops. Our results show that bistable behavior in covalent modification cascades relies not only on the structure and regulatory pattern of feedback/feedforward loops, but also on the kinetic characteristics of their component proteins.     

Energetics of cooperative protein-DNA interactions: comparison between quantitative deoxyribonuclease footprint titration and filter binding

Using the binding of cI repressor protein to the lambda right and left operators as a model system, we have analyzed the two common experimental techniques for studying the interactions of genome regulatory proteins with multiple, specific sites on DNA. These are the quantitative DNase footprint titration technique [Brenowitz, M., Senear, D. F., Shea, M. A., & Ackers, G. K. (1986) Methods Enzymol. 130, 132-181] and the nitrocellulose filter binding assay [Riggs, A., Suzuki, H., & Bourgeois, S. (1970) J. Mol. Biol. 48, 67-83]. The footprint titration technique provides binding curves that separately represent the fractional saturation for each site. In principle, such data contain the information necessary to determine the thermodynamic constants for local site binding and cooperativity. We show that in practice, this is not possible for all values of the constants in multisite systems, such as the lambda operators. We show how these constants can nevertheless be uniquely determined by using additional binding data from a small number of mutant operators in which the number of binding sites has been reduced. The filter binding technique does not distinguish binding to the individual sites and yields only macroscopic binding parameters which are composite averages of the various local site and cooperativity constants. Moreover, the resolution of even macroscopic constants from filter binding data for multisite systems requires ad hoc assumptions as to a relationship between the number of ligands bound and the filter retention of the complex. Our results indicate that no such relationship exists. Hence, the technique does not permit determination of thermodynamically valid interaction constants (even macroscopic) in multisite systems.     

Analysing covariates with spike at zero: A modified FP procedure and conceptual issues

In epidemiology and in clinical research, risk factors often have special distributions. A common situation is that a proportion of individuals have exposure zero, and among those exposed, we have some continuous distribution. We call this a ‘spike at zero’. Examples for this are smoking, duration of breastfeeding, or alcohol consumption. Furthermore, the empirical distribution of laboratory values and other measurements may have a semi‐continuous distribution as a result of the lower detection limit of the measurement. To model the dose–response function, an extension of the fractional polynomial approach was recently proposed. In this paper, we suggest a modification of the previously suggested FP procedure. We first give the theoretical justification of this modified procedure by investigating relevant distribution classes. Here, we systematically derive the theoretical shapes of dose–response curves under given distributional assumptions (normal, log normal, gamma) in the framework of a logistic regression model. Further, we check the performance of the procedure in a simulation study and compare it to the previously suggested method, and finally we illustrate the procedures with data from a case–control study on breast cancer.     

A modular approach for modeling the cell cycle based on functional response curves

Modeling biochemical reactions by means of differential equations often results in systems with a large number of variables and parameters. As this might complicate the interpretation and generalization of the obtained results, it is often desirable to reduce the complexity of the model. One way to accomplish this is by replacing the detailed reaction mechanisms of certain modules in the model by a mathematical expression that qualitatively describes the dynamical behavior of these modules. Such an approach has been widely adopted for ultrasensitive responses, for which underlying reaction mechanisms are often replaced by a single Hill function. Also time delays are usually accounted for by using an explicit delay in delay differential equations. In contrast, however, S-shaped response curves, which by definition have multiple output values for certain input values and are often encountered in bistable systems, are not easily modeled in such an explicit way. Here, we extend the classical Hill function into a mathematical expression that can be used to describe both ultrasensitive and S-shaped responses. We show how three ubiquitous modules (ultrasensitive responses, S-shaped responses and time delays) can be combined in different configurations and explore the dynamics of these systems. As an example, we apply our strategy to set up a model of the cell cycle consisting of multiple bistable switches, which can incorporate events such as DNA damage and coupling to the circadian clock in a phenomenological way.     

Ultrasensitive synthetic protein regulatory networks using mixed decoys

Cellular protein interaction networks exhibit sigmoidal input-output relationships with thresholds and steep responses (i.e. ultrasensitivity). Although cooperativity can be a source of ultrasensitivity, we examined whether the presence of "decoy" binding sites that are not coupled to activation could also lead to this effect. To systematically vary key parameters of the system, we designed a synthetic regulatory system consisting of an autoinhibited PDZ domain coupled to an activating SH3 domain binding site. In the absence of a decoy binding site, this system is non-ultrasensitive, as predicted by modeling of this system. Addition of a high-affinity decoy site adds a threshold, but the response is not ultrasensitive. We found that sigmoidal activation profiles can be generated utilizing multiple decoys with mixtures of high and low affinities, where high affinity decoys act to set the threshold and low affinity decoys ensure a sigmoidal response. Placing the synthetic decoy system in a mitotic spindle orientation cell culture system thresholds this physiological activity. Thus, simple combinations of non-activating binding sites can lead to complex regulatory responses in protein interaction networks.