Machines vs. ensembles: effective MAPK signaling through heterogeneous sets of protein complexes

Modeling reveals the strength of weak interactions in stacked-ring assembly

Cells employ many large macromolecular machines for the execution and regulation of processes that are vital for cell and organismal viability. Interestingly, cells cannot synthesize these machines as functioning units. Instead, cells synthesize the molecular parts that must then assemble into the functional complex. Many important machines, including chaperones such as GroEL and proteases such as the proteasome, comprise protein rings that are stacked on top of one another. While there is some experimental data regarding how stacked-ring complexes such as the proteasome self-assemble, a comprehensive understanding of the dynamics of stacked-ring assembly is currently lacking. Here, we developed a mathematical model of stacked-trimer assembly and performed an analysis of the assembly of the stacked homomeric trimer, which is the simplest stacked-ring architecture. We found that stacked rings are particularly susceptible to a form of kinetic trapping that we term "deadlock," in which the system gets stuck in a state where there are many large intermediates that are not the fully assembled structure but that cannot productively react. When interaction affinities are uniformly strong, deadlock severely limits assembly yield. We thus predicted that stacked rings would avoid situations where all interfaces in the structure have high affinity. Analysis of available crystal structures indicated that indeed the majority-if not all-of stacked trimers do not contain uniformly strong interactions. Finally, to better understand the origins of deadlock, we developed a formal pathway analysis and showed that, when all the binding affinities are strong, many of the possible pathways are utilized. In contrast, optimal assembly strategies utilize only a small number of pathways. Our work suggests that deadlock is a critical factor influencing the evolution of macromolecular machines and provides general principles for understanding the self-assembly efficiency of existing machines.

Design patterns of biological cells

Design patterns are generalized solutions to frequently recurring problems. They were initially developed by architects and computer scientists to create a higher level of abstraction for their designs. Here, we extend these concepts to cell biology to lend a new perspective on the evolved designs of cells' underlying reaction networks. We present a catalog of 21 design patterns divided into three categories: creational patterns describe processes that build the cell, structural patterns describe the layouts of reaction networks, and behavioral patterns describe reaction network function. Applying this pattern language to the E. coli central metabolic reaction network, the yeast pheromone response signaling network, and other examples lends new insights into these systems.

Comparison of rule- and ordinary differential equation-based dynamic model of DARPP-32 signalling network

Dynamic modelling has considerably improved our understanding of complex molecular mechanisms. Ordinary differential equations (ODEs) are the most detailed and popular approach to modelling the dynamics of molecular systems. However, their application in signalling networks, characterised by multi-state molecular complexes, can be prohibitive. Contemporary modelling methods, such as rule- based (RB) modelling, have addressed these issues. The advantages of RB modelling over ODEs have been presented and discussed in numerous reviews. In this study, we conduct a direct comparison of the time courses of a molecular system founded on the same reaction network but encoded in the two frameworks. To make such a comparison, a set of reactions that underlie an ODE model was manually encoded in the Kappa language, one of the RB implementations. A comparison of the models was performed at the level of model specification and dynamics, acquired through model simulations. In line with previous reports, we confirm that the Kappa model recapitulates the general dynamics of its ODE counterpart with minor differences. These occur when molecules have multiple sites binding the same interactor. Furthermore, activation of these molecules in the RB model is slower than in the ODE one. As reported for other molecular systems, we find that, also for the DARPP-32 reaction network, the RB representation offers a more expressive and flexible syntax that facilitates access to fine details of the model, easing model reuse. In parallel with these analyses, we report a refactored model of the DARPP-32 interaction network that can serve as a canvas for the development of more complex dynamic models to study this important molecular system.

Robustness and the evolution of length control strategies in the T3SS and flagellar hook

Bacterial cells construct many structures, such as the flagellar hook and the type III secretion system (T3SS) injectisome, that aid in crucial physiological processes such as locomotion and pathogenesis. Both of these structures involve long extracellular channels, and the length of these channels must be highly regulated in order for these structures to perform their intended functions. There are two leading models for how length control is achieved in the flagellar hook and T3SS needle: the substrate switching model, in which the length is controlled by assembly of an inner rod, and the ruler model, in which a molecular ruler controls the length. Although there is qualitative experimental evidence to support both models, comparatively little has been done to quantitatively characterize these mechanisms or make detailed predictions that could be used to unambiguously test these mechanisms experimentally. In this work, we constructed a mathematical model of length control based on the ruler mechanism and found that the predictions of this model are consistent with experimental data-not just for the scaling of the average length with the ruler protein length, but also for the variance. Interestingly, we found that the ruler mechanism allows for the evolution of needles with large average lengths without the concomitant large increase in variance that occurs in the substrate switching mechanism. In addition to making further predictions that can be tested experimentally, these findings shed new light on the trade-offs that may have led to the evolution of different length control mechanisms in different bacterial species.

Crosstalk and ultrasensitivity in protein degradation pathways

Protein turnover is vital to cellular homeostasis. Many proteins are degraded efficiently only after they have been post-translationally “tagged” with a polyubiquitin chain. Ubiquitylation is a form of Post-Translational Modification (PTM): addition of a ubiquitin to the chain is catalyzed by E3 ligases, and removal of ubiquitin is catalyzed by a De-UBiquitylating enzyme (DUB). Nearly four decades ago, Goldbeter and Koshland discovered that reversible PTM cycles function like on-off switches when the substrates are at saturating concentrations. Although this finding has had profound implications for the understanding of switch-like behavior in biochemical networks, the general behavior of PTM cycles subject to synthesis and degradation has not been studied. Using a mathematical modeling approach, we found that simply introducing protein turnover to a standard modification cycle has profound effects, including significantly reducing the switch-like nature of the response. Our findings suggest that many classic results on PTM cycles may not hold in vivo where protein turnover is ubiquitous. We also found that proteins sharing an E3 ligase can have closely related changes in their expression levels. These results imply that it may be difficult to interpret experimental results obtained from either overexpressing or knocking down protein levels, since changes in protein expression can be coupled via E3 ligase crosstalk. Understanding crosstalk and competition for E3 ligases will be key in ultimately developing a global picture of protein homeostasis.

Previous work has shown that substrates of Post-Translational Modification (PTM) cycles can have coupled responses if those substrates share enzymes. This implies that modifications leading to substrate degradation (e.g. ubiquitylation by an E3 ligase) could introduce coupling in concentrations of substrates sharing a ligase. Using mathematical models, we found adding protein turnover to a PTM cycle diminishes both sensitivity and ultrasensitivity, particularly in models admitting long ubiquitin chains. We also found that proteins sharing an E3 ligase can indeed have coupled changes in both expression and sensitivity to signals. These results imply that accounting for crosstalk in protein degradation networks is crucial for the interpretation of results from a wide variety of common experimental perturbations to living systems.

Understand the Functions of Scaffold Proteins in Cell Signaling by a Mesoscopic Simulation Method

Scaffold proteins are central players in regulating the spatial-temporal organization of many important signaling pathways in cells. They offer physical platforms to downstream signaling proteins so that their transient interactions in a crowded and heterogeneous environment of cytosol can be greatly facilitated. However, most scaffold proteins tend to simultaneously bind more than one signaling molecule, which leads to the spatial assembly of multimeric protein complexes. The kinetics of these protein oligomerizations are difficult to quantify by traditional experimental approaches. To understand the functions of scaffold proteins in cell signaling, we developed a, to our knowledge, new hybrid simulation algorithm in which both spatial organization and binding kinetics of proteins were implemented. We applied this new technique to a simple network system that contains three molecules. One molecule in the network is a scaffold protein, whereas the other two are its binding targets in the downstream signaling pathway. Each of the three molecules in the system contains two binding motifs that can interact with each other and are connected by a flexible linker. By applying the new simulation method to the model, we show that the scaffold proteins will promote not only thermodynamics but also kinetics of cell signaling given the premise that the interaction between the two signaling molecules is transient. Moreover, by changing the flexibility of the linker between two binding motifs, our results suggest that the conformational fluctuations in a scaffold protein play a positive role in recruiting downstream signaling molecules. In summary, this study showcases the capability of computational simulation in understanding the general principles of scaffold protein functions.

Combinatorial protein-protein interactions on a polymerizing scaffold

Scaffold proteins organize cellular processes by bringing signaling molecules into interaction, sometimes by forming large signalosomes. Several of these scaffolds are known to polymerize. Their assemblies should therefore not be understood as stoichiometric aggregates, but as combinatorial ensembles. We analyze the combinatorial interaction of ligands loaded on polymeric scaffolds, in both a continuum and discrete setting, and compare it with multivalent scaffolds with fixed number of binding sites. The quantity of interest is the abundance of ligand interaction possibilities-the catalytic potential Q-in a configurational mixture. Upon increasing scaffold abundance, scaffolding systems are known to first increase opportunities for ligand interaction and then to shut them down as ligands become isolated on distinct scaffolds. The polymerizing system stands out in that the dependency of Q on protomer concentration switches from being dominated by a first order to a second order term within a range determined by the polymerization affinity. This behavior boosts Q beyond that of any multivalent scaffold system. In addition, the subsequent drop-off is considerably mitigated in that Q decreases with half the power in protomer concentration than for any multivalent scaffold. We explain this behavior in terms of how the concentration profile of the polymer-length distribution adjusts to changes in protomer concentration and affinity. The discrete case turns out to be similar, but the behavior can be exaggerated at small protomer numbers because of a maximal polymer size, analogous to finite-size effects in bond percolation on a lattice.

PyBioNetFit and the Biological Property Specification Language

In systems biology modeling, important steps include model parameterization, uncertainty quantification, and evaluation of agreement with experimental observations. To help modelers perform these steps, we developed the software PyBioNetFit, which in addition supports checking models against known system properties and solving design problems. PyBioNetFit introduces Biological Property Specification Language (BPSL) for the formal declaration of system properties. BPSL allows qualitative data to be used alone or in combination with quantitative data. PyBioNetFit performs parameterization with parallelized metaheuristic optimization algorithms that work directly with existing model definition standards: BioNetGen Language (BNGL) and Systems Biology Markup Language (SBML). We demonstrate PyBioNetFit's capabilities by solving various example problems, including the challenging problem of parameterizing a 153-parameter model of cell cycle control in yeast based on both quantitative and qualitative data. We demonstrate the model checking and design applications of PyBioNetFit and BPSL by analyzing a model of targeted drug interventions in autophagy signaling.

Rule-Based Modeling Using Wildcards in the Smoldyn Simulator

Many biological molecules exist in multiple variants, such as proteins with different posttranslational modifications, DNAs with different sequences, and phospholipids with different chain lengths. Representing these variants as distinct species, as most biochemical simulators do, leads to the problem that the number of species, and chemical reactions that interconvert them, typically increase combinatorially with the number of ways that the molecules can vary. This can be alleviated by "rule-based modeling methods," in which software generates the chemical reaction network from relatively simple "rules." This chapter presents a new approach to rule-based modeling. It is based on wildcards that match to species names, much as wildcards can match to file names in computer operating systems. It is much simpler to use than the formal rule-based modeling approaches developed previously but can lead to unintended consequences if not used carefully. This chapter demonstrates rule-based modeling with wildcards through examples for signaling systems, protein complexation, polymerization, nucleic acid sequence copying and mutation, the "SMILES" chemical notation, and others. The method is implemented in Smoldyn, a spatial and stochastic biochemical simulator, for both generate-first and on-the-fly expansion, meaning whether the reaction network is generated before or during the simulation.

Is the cell really a machine?

It has become customary to conceptualize the living cell as an intricate piece of machinery, different to a man-made machine only in terms of its superior complexity. This familiar understanding grounds the conviction that a cell's organization can be explained reductionistically, as well as the idea that its molecular pathways can be construed as deterministic circuits. The machine conception of the cell owes a great deal of its success to the methods traditionally used in molecular biology. However, the recent introduction of novel experimental techniques capable of tracking individual molecules within cells in real time is leading to the rapid accumulation of data that are inconsistent with an engineering view of the cell. This paper examines four major domains of current research in which the challenges to the machine conception of the cell are particularly pronounced: cellular architecture, protein complexes, intracellular transport, and cellular behaviour. It argues that a new theoretical understanding of the cell is emerging from the study of these phenomena which emphasizes the dynamic, self-organizing nature of its constitution, the fluidity and plasticity of its components, and the stochasticity and non-linearity of its underlying processes.

Computational approaches to macromolecular interactions in the cell

Structural modeling of a cell is an evolving strategic direction in computational structural biology. It takes advantage of new powerful modeling techniques, deeper understanding of fundamental principles of molecular structure and assembly, and rapid growth of the amount of structural data generated by experimental techniques. Key modeling approaches to principal types of macromolecular assemblies in a cell already exist. The main challenge, along with the further development of these modeling approaches, is putting them together in a consistent, unified whole cell model. This opinion piece addresses the fundamental aspects of modeling macromolecular assemblies in a cell, and the state-of-the-art in modeling of the principal types of such assemblies.

Using RuleBuilder to Graphically Define and Visualize BioNetGen-Language Patterns and Reaction Rules

RuleBuilder is a tool for drawing graphs that can be represented by the BioNetGen language (BNGL), which is used to formulate mathematical, rule-based models of biochemical systems. BNGL provides an intuitive plain text, or string, representation of such systems, which is based on a graphical formalism. Reactions are defined in terms of graph-rewriting rules that specify the necessary intrinsic properties of the reactants, a transformation, and a rate law. Rules also contain contextual constraints that restrict application of the rule. In some cases, the specification of contextual constraints can be verbose, making a rule difficult to read. RuleBuilder is designed to ease the task of reading and writing individual reaction rules or other BNGL patterns required for model formulation. The software assists in the reading of existing models by converting BNGL strings of interest into a graph-based representation composed of nodes and edges. RuleBuilder also enables the user to construct de novo a visual representation of BNGL strings using drawing tools available in its interface. As objects are added to the drawing canvas, the corresponding BNGL string is generated on the fly, and objects are similarly drawn on the fly as BNGL strings are entered into the application. RuleBuilder thus facilitates construction and interpretation of rule-based models.

PyDREAM: high-dimensional parameter inference for biological models in python

Summary

Biological models contain many parameters whose values are difficult to measure directly via experimentation and therefore require calibration against experimental data. Markov chain Monte Carlo (MCMC) methods are suitable to estimate multivariate posterior model parameter distributions, but these methods may exhibit slow or premature convergence in high-dimensional search spaces. Here, we present PyDREAM, a Python implementation of the (Multiple-Try) Differential Evolution Adaptive Metropolis [DREAM_(ZS)] algorithm developed by Vrugt and ter Braak (2008) and Laloy and Vrugt (2012). PyDREAM achieves excellent performance for complex, parameter-rich models and takes full advantage of distributed computing resources, facilitating parameter inference and uncertainty estimation of CPU-intensive biological models.

Availability and implementation

PyDREAM is freely available under the GNU GPLv3 license from the Lopez lab GitHub repository at http://github.com/LoLab-VU/PyDREAM.

Supplementary information

Supplementary data are available at Bioinformatics online.

The Interplay of Structural and Cellular Biophysics Controls Clustering of Multivalent Molecules

Dynamic molecular clusters are assembled through weak multivalent interactions and are platforms for cellular functions, especially receptor-mediated signaling. Clustering is also a prerequisite for liquid-liquid phase separation. It is not well understood, however, how molecular structure and cellular organization control clustering. Using coarse-grained kinetic Langevin dynamics, we performed computational experiments on a prototypical ternary system modeled after membrane-bound nephrin, the adaptor Nck1, and the actin nucleation promoting factor NWASP. Steady-state cluster size distributions favored stoichiometries that optimized binding (stoichiometry matching) but still were quite broad. At high concentrations, the system can be driven beyond the saturation boundary such that cluster size is limited only by the number of available molecules. This behavior would be predictive of phase separation. Domains close to binding sites sterically inhibited clustering much less than terminal domains because the latter effectively restrict access to the cluster interior. Increased flexibility of interacting molecules diminished clustering by shielding binding sites within compact conformations. Membrane association of nephrin increased the cluster size distribution in a density-dependent manner. These properties provide insights into how molecular ensembles function to localize and amplify cell signaling.

Intrinsic limits of information transmission in biochemical signalling motifs

All living things have evolved to sense changes in their environment in order to respond in adaptive ways. At the cellular level, these sensing systems generally involve receptor molecules at the cell surface, which detect changes outside the cell and relay those changes to the appropriate response elements downstream. With the advent of experimental technologies that can track signalling at the single-cell level, it has become clear that many signalling systems exhibit significant levels of 'noise,' manifesting as differential responses of otherwise identical cells to the same environment. This noise has a large impact on the capacity of cell signalling networks to transmit information from the environment. Application of information theory to experimental data has found that all systems studied to date encode less than 2.5 bits of information, with the majority transmitting significantly less than 1 bit. Given the growing interest in applying information theory to biological data, it is crucial to understand whether the low values observed to date represent some sort of intrinsic limit on information flow given the inherently stochastic nature of biochemical signalling events. In this work, we used a series of computational models to explore how much information a variety of common 'signalling motifs' can encode. We found that the majority of these motifs, which serve as the basic building blocks of cell signalling networks, can encode far more information (4-6 bits) than has ever been observed experimentally. In addition to providing a consistent framework for estimating information-theoretic quantities from experimental data, our findings suggest that the low levels of information flow observed so far in living system are not necessarily due to intrinsic limitations. Further experimental work will be needed to understand whether certain cell signalling systems actually can approach the intrinsic limits described here, and to understand the sources and purpose of the variation that reduces information flow in living cells.

GPU-powered model analysis with PySB/cupSODA

Summary

A major barrier to the practical utilization of large, complex models of biochemical systems is the lack of open-source computational tools to evaluate model behaviors over high-dimensional parameter spaces. This is due to the high computational expense of performing thousands to millions of model simulations required for statistical analysis. To address this need, we have implemented a user-friendly interface between cupSODA, a GPU-powered kinetic simulator, and PySB, a Python-based modeling and simulation framework. For three example models of varying size, we show that for large numbers of simulations PySB/cupSODA achieves order-of-magnitude speedups relative to a CPU-based ordinary differential equation integrator.

Availability and implementation

The PySB/cupSODA interface has been integrated into the PySB modeling framework (version 1.4.0), which can be installed from the Python Package Index (PyPI) using a Python package manager such as pip. cupSODA source code and precompiled binaries (Linux, Mac OS/X, Windows) are available at github.com/aresio/cupSODA (requires an Nvidia GPU; developer.nvidia.com/cuda-gpus). Additional information about PySB is available at pysb.org.

Contact

paolo.cazzaniga@unibg.it or c.lopez@vanderbilt.edu.

Supplementary information

Supplementary data are available at Bioinformatics online.

Generalizing Gillespie's Direct Method to Enable Network-Free Simulations

Gillespie's direct method for stochastic simulation of chemical kinetics is a staple of computational systems biology research. However, the algorithm requires explicit enumeration of all reactions and all chemical species that may arise in the system. In many cases, this is not feasible due to the combinatorial explosion of reactions and species in biological networks. Rule-based modeling frameworks provide a way to exactly represent networks containing such combinatorial complexity, and generalizations of Gillespie's direct method have been developed as simulation engines for rule-based modeling languages. Here, we provide both a high-level description of the algorithms underlying the simulation engines, termed network-free simulation algorithms, and how they have been applied in systems biology research. We also define a generic rule-based modeling framework and describe a number of technical details required for adapting Gillespie's direct method for network-free simulation. Finally, we briefly discuss potential avenues for advancing network-free simulation and the role they continue to play in modeling dynamical systems in biology.

Toward Large-Scale Computational Prediction of Protein Complexes

Cellular functions are often performed by multiprotein structures called protein complexes. These complexes are dynamic structures that evolve during the cell cycle or in response to external and internal stimuli, and are tightly regulated by protein expression in different tissues resulting in quantitative and qualitative variation of protein complexes. Advances in high-throughput techniques, such as mass-spectrometry and yeast two-hybrid provided a large amount of data on protein-protein interactions. This sparked the development of computational methods able to predict protein complex formation under a variety of biological and clinical conditions. However, the challenges that need to be addressed for successful computational protein complex prediction are highly complex.The post-genomic era saw an emerging number of algorithms and software, which are able to predict protein complexes from protein-protein interaction networks and a variety of other sources. Despite the high capacity of these methods to qualitatively predict protein complexes, they could provide only limited or no quantitative information of the predicted complexes. Recently, a new large-scale simulation of protein complexes was able to achieve this task by simulating protein complex formation on the proteome scale.In this chapter, we review representative methods that can predict multiple protein complexes at different scales and discuss how these can be combined with emerging sources of data in order to improve protein complex characterization.

Automated visualization of rule-based models

Frameworks such as BioNetGen, Kappa and Simmune use "reaction rules" to specify biochemical interactions compactly, where each rule specifies a mechanism such as binding or phosphorylation and its structural requirements. Current rule-based models of signaling pathways have tens to hundreds of rules, and these numbers are expected to increase as more molecule types and pathways are added. Visual representations are critical for conveying rule-based models, but current approaches to show rules and interactions between rules scale poorly with model size. Also, inferring design motifs that emerge from biochemical interactions is an open problem, so current approaches to visualize model architecture rely on manual interpretation of the model. Here, we present three new visualization tools that constitute an automated visualization framework for rule-based models: (i) a compact rule visualization that efficiently displays each rule, (ii) the atom-rule graph that conveys regulatory interactions in the model as a bipartite network, and (iii) a tunable compression pipeline that incorporates expert knowledge and produces compact diagrams of model architecture when applied to the atom-rule graph. The compressed graphs convey network motifs and architectural features useful for understanding both small and large rule-based models, as we show by application to specific examples. Our tools also produce more readable diagrams than current approaches, as we show by comparing visualizations of 27 published models using standard graph metrics. We provide an implementation in the open source and freely available BioNetGen framework, but the underlying methods are general and can be applied to rule-based models from the Kappa and Simmune frameworks also. We expect that these tools will promote communication and analysis of rule-based models and their eventual integration into comprehensive whole-cell models.

A DNA nanoscope via auto-cycling proximity recording

Analysis of the spatial arrangement of molecular features enables the engineering of synthetic nanostructures and the understanding of natural ones. The ability to acquire a comprehensive set of pairwise proximities between components would satisfy an increasing interest in investigating individual macromolecules and their interactions, but current biochemical techniques detect only a single proximity partner per probe. Here, we present a biochemical DNA nanoscopy method that records nanostructure features in situ and in detail for later readout. Based on a conceptually novel auto-cycling proximity recording (APR) mechanism, it continuously and repeatedly produces proximity records of any nearby pairs of DNA-barcoded probes, at physiological temperature, without altering the probes themselves. We demonstrate the production of dozens of records per probe, decode the spatial arrangements of 7 unique probes in a homogeneous sample, and repeatedly sample the same probes in different states.

The spatial organisation of nanostructures is fundamental to their function. Here, the authors develop a non-destructive, proximity-based method to record extensive spatial organization information in DNA molecules for later readout.

Fundamental trade-offs between information flow in single cells and cellular populations

Signal transduction networks allow eukaryotic cells to make decisions based on information about intracellular state and the environment. Biochemical noise significantly diminishes the fidelity of signaling: networks examined to date seem to transmit less than 1 bit of information. It is unclear how networks that control critical cell-fate decisions (e.g., cell division and apoptosis) can function with such low levels of information transfer. Here, we use theory, experiments, and numerical analysis to demonstrate an inherent trade-off between the information transferred in individual cells and the information available to control population-level responses. Noise in receptor-mediated apoptosis reduces information transfer to approximately 1 bit at the single-cell level but allows 3-4 bits of information to be transmitted at the population level. For processes such as eukaryotic chemotaxis, in which single cells are the functional unit, we find high levels of information transmission at a single-cell level. Thus, low levels of information transfer are unlikely to represent a physical limit. Instead, we propose that signaling networks exploit noise at the single-cell level to increase population-level information transfer, allowing extracellular ligands, whose levels are also subject to noise, to incrementally regulate phenotypic changes. This is particularly critical for discrete changes in fate (e.g., life vs. death) for which the key variable is the fraction of cells engaged. Our findings provide a framework for rationalizing the high levels of noise in metazoan signaling networks and have implications for the development of drugs that target these networks in the treatment of cancer and other diseases.

Introduction of Premature Stop Codons as an Evolutionary Strategy To Rescue Signaling Network Function

The cellular concentrations of key components of signaling networks are tightly regulated, as deviations from their optimal ranges can have negative effects on signaling function. For example, overexpression of the yeast mating pathway mitogen-activated protein kinase (MAPK) Fus3 decreases pathway output, in part by sequestering individual components away from functional multiprotein complexes. Using a synthetic biology approach, we investigated potential mechanisms by which selection could compensate for a decrease in signaling activity caused by overexpression of Fus3. We overexpressed a library of random mutants of Fus3 and used cell sorting to select variants that rescued mating pathway activity. Our results uncovered that one remarkable way in which selection can compensate for protein overexpression is by introducing premature stop codons at permitted positions. Because of the low efficiency with which premature stop codons are read through, the resulting cellular concentration of active Fus3 returns to values within the range required for proper signaling. Our results underscore the importance of interpreting genotypic variation at the systems rather than at the individual gene level, as mutations can have opposite effects on protein and network function.

Multivalent Molecules as Modulators of RNA Granule Size and Composition

RNA granules are ensembles of specific RNA and protein molecules that mediate localized translation in eukaryotic cells. The mechanisms for formation and selectivity of RNA granules are unknown. Here we present a model for assembly of one type of RNA granule based on experimentally measured binding interactions among three core multivalent molecular components necessary for such assembly: specific RNA molecules that contain a cis-acting sequence called the A2 response element (A2RE), hnRNP A2 proteins that bind specifically (with high affinity) to A2RE sequences or nonspecifically (with lower affinity) to other RNA sequences, and heptavalent protein cytoskeleton-associated protein 5 (CKAP5, an alternative name for TOG protein) that binds both hnRNP A2 molecules and RNA. Non-A2RE RNA molecules (RNA without the A2RE sequence) that may be recruited to the granules through nonspecific interactions are also considered in the model. Modeling multivalent molecular interactions in granules is challenging because of combinatorial complexity in the number of potential molecular complexes among these core components and dynamic changes in granule composition and structure in response to changes in local intracellular environment. We use a hybrid modeling approach (deterministic-stochastic-statistical) that is appropriate when the overall compositions of multimolecular ensembles are of greater importance than the specific interactions among individual molecular components. Modeling studies titrating the concentrations of various granule components and varying effective site pair affinities and RNA valency demonstrate that interactions between multivalent components (TOG and RNA) are modulated by a bivalent adaptor molecule (hnRNP A2). Formation and disruption of granules, as well as RNA selectivity in granule composition are regulated by distinct concentration regimes of A2. Our results suggest that granule assembly is tightly controlled by multivalent molecular interactions among RNA molecules, adaptor proteins, and scaffold proteins.

SpringSaLaD: A Spatial, Particle-Based Biochemical Simulation Platform with Excluded Volume

We introduce Springs, Sites, and Langevin Dynamics (SpringSaLaD), a comprehensive software platform for spatial, stochastic, particle-based modeling of biochemical systems. SpringSaLaD models biomolecules in a coarse-grained manner as a group of linked spherical sites with excluded volume. This mesoscopic approach bridges the gap between highly detailed molecular dynamics simulations and the various methods used to study network kinetics and diffusion at the cellular level. SpringSaLaD is a standalone tool that supports model building, simulation, visualization, and data analysis, all through a user-friendly graphical user interface that should make it more accessible than tools built into more comprehensive molecular dynamics infrastructures. Importantly, for bimolecular reactions we derive an exact expression relating the macroscopic on-rate to the various microscopic parameters with the inclusion of excluded volume; this makes SpringSaLaD more accurate than other tools, which rely on approximate relationships between these parameters.

Protein Ensembles: How Does Nature Harness Thermodynamic Fluctuations for Life? The Diverse Functional Roles of Conformational Ensembles in the Cell

All soluble proteins populate conformational ensembles that together constitute the native state. Their fluctuations in water are intrinsic thermodynamic phenomena, and the distributions of the states on the energy landscape are determined by statistical thermodynamics; however, they are optimized to perform their biological functions. In this review we briefly describe advances in free energy landscape studies of protein conformational ensembles. Experimental (nuclear magnetic resonance, small-angle X-ray scattering, single-molecule spectroscopy, and cryo-electron microscopy) and computational (replica-exchange molecular dynamics, metadynamics, and Markov state models) approaches have made great progress in recent years. These address the challenging characterization of the highly flexible and heterogeneous protein ensembles. We focus on structural aspects of protein conformational distributions, from collective motions of single- and multi-domain proteins, intrinsically disordered proteins, to multiprotein complexes. Importantly, we highlight recent studies that illustrate functional adjustment of protein conformational ensembles in the crowded cellular environment. We center on the role of the ensemble in recognition of small- and macro-molecules (protein and RNA/DNA) and emphasize emerging concepts of protein dynamics in enzyme catalysis. Overall, protein ensembles link fundamental physicochemical principles and protein behavior and the cellular network and its regulation.

Symbolic computation of differential equivalences

Ordinary differential equations (ODEs) are widespread in many natural sciences including chemistry, ecology, and systems biology, and in disciplines such as control theory and electrical engineering. Building on the celebrated molecules-as-processes paradigm, they have become increasingly popular in computer science, with high-level languages and formal methods such as Petri nets, process algebra, and rule-based systems that are interpreted as ODEs. We consider the problem of comparing and minimizing ODEs automatically. Influenced by traditional approaches in the theory of programming, we propose differential equivalence relations. We study them for a basic intermediate language, for which we have decidability results, that can be targeted by a class of high-level specifications. An ODE implicitly represents an uncountable state space, hence reasoning techniques cannot be borrowed from established domains such as probabilistic programs with finite-state Markov chain semantics. We provide novel symbolic procedures to check an equivalence and compute the largest one via partition refinement algorithms that use satisfiability modulo theories. We illustrate the generality of our framework by showing that differential equivalences include (i) well-known notions for the minimization of continuous-time Markov chains (lumpability), (ii)~bisimulations for chemical reaction networks recently proposed by Cardelli et al., and (iii) behavioral relations for process algebra with ODE semantics. With a prototype implementation we are able to detect equivalences in biochemical models from the literature that cannot be reduced using competing automatic techniques.

Mathematical Model for Length Control by the Timing of Substrate Switching in the Type III Secretion System

Type III Secretion Systems (T3SS) are complex bacterial structures that provide gram-negative pathogens with a unique virulence mechanism whereby they grow a needle-like structure in order to inject bacterial effector proteins into the cytoplasm of a host cell. Numerous experiments have been performed to understand the structural details of this nanomachine during the past decade. Despite the concerted efforts of molecular and structural biologists, several crucial aspects of the assembly of this structure, such as the regulation of the length of the needle itself, remain unclear. In this work, we used a combination of mathematical and computational techniques to better understand length control based on the timing of substrate switching, which is a possible mechanism for how bacteria ensure that the T3SS needles are neither too short nor too long. In particular, we predicted the form of the needle length distribution based on this mechanism, and found excellent agreement with available experimental data from Salmonella typhimurium with only a single free parameter. Although our findings provide preliminary evidence in support of the substrate switching model, they also make a set of quantitative predictions that, if tested experimentally, would assist in efforts to unambiguously characterize the regulatory mechanisms that control the growth of this crucial virulence factor.

Phosphatase specificity and pathway insulation in signaling networks

Phosphatases play an important role in cellular signaling networks by regulating the phosphorylation state of proteins. Phosphatases are classically considered to be promiscuous, acting on tens to hundreds of different substrates. We recently demonstrated that a shared phosphatase can couple the responses of two proteins to incoming signals, even if those two substrates are from otherwise isolated areas of the network. This finding raises a potential paradox: if phosphatases are indeed highly promiscuous, how do cells insulate themselves against unwanted crosstalk? Here, we use mathematical models to explore three possible insulation mechanisms. One approach involves evolving phosphatase KM values that are large enough to prevent saturation by the phosphatase's substrates. Although this is an effective method for generating isolation, the phosphatase becomes a highly inefficient enzyme, which prevents the system from achieving switch-like responses and can result in slow response kinetics. We also explore the idea that substrate degradation can serve as an effective phosphatase. Assuming that degradation is unsaturatable, this mechanism could insulate substrates from crosstalk, but it would also preclude ultrasensitive responses and would require very high substrate turnover to achieve rapid dephosphorylation kinetics. Finally, we show that adaptor subunits, such as those found on phosphatases like PP2A, can provide effective insulation against phosphatase crosstalk, but only if their binding to substrates is uncoupled from their binding to the catalytic core. Analysis of the interaction network of PP2A's adaptor domains reveals that although its adaptors may isolate subsets of targets from one another, there is still a strong potential for phosphatase crosstalk within those subsets. Understanding how phosphatase crosstalk and the insulation mechanisms described here impact the function and evolution of signaling networks represents a major challenge for experimental and computational systems biology.

Evolution of synthetic signaling scaffolds by recombination of modular protein domains

Signaling scaffolds are proteins that interact via modular domains with multiple partners, regulating signaling networks in space and time and providing an ideal platform from which to alter signaling functions. However, to better exploit scaffolds for signaling engineering, it is necessary to understand the full extent of their modularity. We used a directed evolution approach to identify, from a large library of randomly shuffled protein interaction domains, variants capable of rescuing the signaling defect of a yeast strain in which Ste5, the scaffold in the mating pathway, had been deleted. After a single round of selection, we identified multiple synthetic scaffold variants with diverse domain architectures, able to mediate mating pathway activation in a pheromone-dependent manner. The facility with which this signaling network accommodates changes in scaffold architecture suggests that the mating signaling complex does not possess a single, precisely defined geometry into which the scaffold has to fit. These relaxed geometric constraints may facilitate the evolution of signaling networks, as well as their engineering for applications in synthetic biology.

Dissecting the calcium-induced differentiation of human primary keratinocytes stem cells by integrative and structural network analyses

The molecular details underlying the time-dependent assembly of protein complexes in cellular networks, such as those that occur during differentiation, are largely unexplored. Focusing on the calcium-induced differentiation of primary human keratinocytes as a model system for a major cellular reorganization process, we look at the expression of genes whose products are involved in manually-annotated protein complexes. Clustering analyses revealed only moderate co-expression of functionally related proteins during differentiation. However, when we looked at protein complexes, we found that the majority (55%) are composed of non-dynamic and dynamic gene products ('di-chromatic'), 19% are non-dynamic, and 26% only dynamic. Considering three-dimensional protein structures to predict steric interactions, we found that proteins encoded by dynamic genes frequently interact with a common non-dynamic protein in a mutually exclusive fashion. This suggests that during differentiation, complex assemblies may also change through variation in the abundance of proteins that compete for binding to common proteins as found in some cases for paralogous proteins. Considering the example of the TNF-α/NFκB signaling complex, we suggest that the same core complex can guide signals into diverse context-specific outputs by addition of time specific expressed subunits, while keeping other cellular functions constant. Thus, our analysis provides evidence that complex assembly with stable core components and competition could contribute to cell differentiation.

The robustness of a signaling complex to domain rearrangements facilitates network evolution

The broad tolerance of domain-rearranging mutations by a yeast signaling network suggests that signaling complexes have loose spatial constraints, making manipulation and perhaps evolution easier.

The rearrangement of protein domains is known to have key roles in the evolution of signaling networks and, consequently, is a major tool used to synthetically rewire networks. However, natural mutational events leading to the creation of proteins with novel domain combinations, such as in frame fusions followed by domain loss, retrotranspositions, or translocations, to name a few, often simultaneously replace pre-existing genes. Thus, while proteins with new domain combinations may establish novel network connections, it is not clear how the concomitant deletions are tolerated. We investigated the mechanisms that enable signaling networks to tolerate domain rearrangement-mediated gene replacements. Using as a model system the yeast mitogen activated protein kinase (MAPK)-mediated mating pathway, we analyzed 92 domain-rearrangement events affecting 11 genes. Our results indicate that, while domain rearrangement events that result in the loss of catalytic activities within the signaling complex are not tolerated, domain rearrangements can drastically alter protein interactions without impairing function. This suggests that signaling complexes can maintain function even when some components are recruited to alternative sites within the complex. Furthermore, we also found that the ability of the complex to tolerate changes in interaction partners does not depend on long disordered linkers that often connect domains. Taken together, our results suggest that some signaling complexes are dynamic ensembles with loose spatial constraints that could be easily re-shaped by evolution and, therefore, are ideal targets for cellular engineering.

Cells use complex protein interaction networks to sense and process external signals. Proteins involved in signaling are often composed of multiple functional units called domains. Because domains are modular, mutations that rearrange domains among proteins have the potential to result in the creation of novel proteins with altered functions. At an evolutionary timescale, domain rearrangements contribute to the functional diversification of signaling networks; at the shorter timescale of the life of an individual, domain rearrangements can impair cellular functions and lead to disease. Here, we investigated how domain-rearranging mutations alter the function of signaling networks, in particular when these mutations disrupt pre-existing proteins. We used as a model system the yeast mating signaling pathway, which shares many properties with more complex pathways active in human cells. Our results demonstrate that signaling networks are often robust to domain rearrangements that disrupt pre-existing genes. In addition, our experiments suggest a possible mechanism to explain this robustness: rather than being a rigid multi-protein machine, the yeast mating signaling complex is a dynamic ensemble with loose spatial constraints. Because of this, the changes in protein interaction partners caused by domain-rearrangement mutations can be accommodated without disrupting network function.

Reconstitution of multivalent PDZ domain binding to the scaffold protein PSD-95 reveals ternary-complex specificity of combinatorial inhibition

Multidomain scaffold proteins serve as hubs in the signal transduction network. By physically colocalizing sequential steps in a transduction pathway, scaffolds catalyze and direct incoming signals. Much is known about binary interactions with individual domains, but it is unknown whether "scaffolding activity" is predictable from pairwise affinities. Here, we characterized multivalent binding to PSD-95, a scaffold protein containing three PDZ domains connected in series by disordered linkers. We used single molecule fluorescence to watch soluble PSD-95 recruit diffusing proteins to a surface-attached receptor cytoplasmic domain. Different ternary complexes showed unique concentration dependence for scaffolding despite similar pairwise affinity. The concentration dependence of scaffolding activity was not predictable based on binary interactions. PSD-95 did not stabilize specific complexes, but rather increased the frequency of transient binding events. Our results suggest that PSD-95 maintains a loosely connected pleomorphic ensemble rather than forming a stereospecific complex containing all components.