HYPK: A marginally disordered protein sensitive to charge decoration.

Intrinsically disordered proteins (IDPs) that lie close to the empirical boundary separating IDPs and folded proteins in Uversky's charge-hydropathy plot may behave as "marginal IDPs" and sensitively switch conformation upon changes in environment (temperature, crowding, and charge screening), sequence, or both. In our search for such a marginal IDP, we selected Huntingtin-interacting protein K (HYPK) near that boundary as a candidate; PKIα, also near that boundary, has lower secondary structure propensity; and Crk1, just across the boundary on the folded side, has higher secondary structure propensity. We used a qualitative Förster resonance energy transfer-based assay together with circular dichroism to simultaneously probe global and local conformation. HYPK shows several unique features indicating marginality: a cooperative transition in end-to-end distance with temperature, like Crk1 and folded proteins, but unlike PKIα; enhanced secondary structure upon crowding, in contrast to Crk1 and PKIα; and a cross-over from salt-induced expansion to compaction at high temperature, likely due to a structure-to-disorder transition not seen in Crk1 and PKIα. We then tested HYPK's sensitivity to charge patterning by designing charge-flipped variants including two specific sequences with identical amino acid composition that markedly differ in their predicted size and response to salt. The experimentally observed trends, also including mutants of PKIα, verify the predictions from sequence charge decoration metrics. Marginal proteins like HYPK show features of both folded and disordered proteins that make them sensitive to physicochemical perturbations and structural control by charge patterning.

MaxCal can infer models from coupled stochastic trajectories of gene expression and cell division.

Gene expression is inherently noisy due to small numbers of proteins and nucleic acids inside a cell. Likewise, cell division is stochastic, particularly when tracking at the level of a single cell. The two can be coupled when gene expression affects the rate of cell division. Single-cell time-lapse experiments can measure both fluctuations by simultaneously recording protein levels inside a cell and its stochastic division. These information-rich noisy trajectory data sets can be harnessed to learn about the underlying molecular and cellular details that are often not known a priori. A critical question is: How can we infer a model given data where fluctuations at two levels-gene expression and cell division-are intricately convoluted? We show the principle of maximum caliber (MaxCal)-integrated within a Bayesian framework-can be used to infer several cellular and molecular details (division rates, protein production, and degradation rates) from these coupled stochastic trajectories (CSTs). We demonstrate this proof of concept using synthetic data generated from a known model. An additional challenge in data analysis is that trajectories are often not in protein numbers, but in noisy fluorescence that depends on protein number in a probabilistic manner. We again show that MaxCal can infer important molecular and cellular rates even when data are in fluorescence, another example of CST with three confounding factors-gene expression noise, cell division noise, and fluorescence distortion-all coupled. Our approach will provide guidance to build models in synthetic biology experiments as well as general biological systems where examples of CSTs are abundant.

Intrachain interaction topology can identify functionally similar intrinsically disordered proteins.

Functionally similar IDPs (intrinsically disordered proteins) often have little sequence similarity. This is in stark contrast to folded proteins and poses a challenge for the inverse problem, functional classification of IDPs using sequence alignment. The problem is further compounded because of the lack of structure in IDPs, preventing structural alignment as an alternate tool for classification. Recent advances in heteropolymer theory unveiled a powerful set of sequence-patterning metrics bridging molecular interaction with chain conformation. Focusing only on charge patterning, these set of metrics yield a sequence charge decoration matrix (SCDM). SCDMs can potentially identify functionally similar IDPs not apparent from sequence alignment alone. Here, we illustrate how these information-rich "molecular blueprints" encoded in SCDMs can be used for functional classification of IDPs with specific application in three protein families-Ste50, PSC, and RAM-in which electrostatics is known to be important. For both the Ste50 and PSC protein family, the set of metrics appropriately classifies proteins in functional and nonfunctional groups in agreement with experiment. Furthermore, our algorithm groups synthetic variants of the disordered RAM region of the Notch receptor protein-important in gene expression-in reasonable accordance with classification based on experimentally measured binding constants of RAM and transcription factor. Taken together, the novel classification scheme reveals the critical role of a high-dimensional set of metrics-manifest in self-interaction maps and topology-in functional annotation of IDPs even when there is low sequence homology, providing the much-needed alternate to a traditional sequence alignment tool.

The Maximum Caliber Variational Principle for Nonequilibria.

Ever since Clausius in 1865 and Boltzmann in 1877, the concepts of entropy and of its maximization have been the foundations for predicting how material equilibria derive from microscopic properties. But, despite much work, there has been no equally satisfactory general variational principle for nonequilibrium situations. However, in 1980, a new avenue was opened by E.T. Jaynes and by Shore and Johnson. We review here maximum caliber, which is a maximum-entropy-like principle that can infer distributions of flows over pathways, given dynamical constraints. This approach is providing new insights, particularly into few-particle complex systems, such as gene circuits, protein conformational reaction coordinates, network traffic, bird flocking, cell motility, and neuronal firing.

Critical Comparison of MaxCal and Other Stochastic Modeling Approaches in Analysis of Gene Networks.

Learning the underlying details of a gene network with feedback is critical in designing new synthetic circuits. Yet, quantitative characterization of these circuits remains limited. This is due to the fact that experiments can only measure partial information from which the details of the circuit must be inferred. One potentially useful avenue is to harness hidden information from single-cell stochastic gene expression time trajectories measured for long periods of time-recorded at frequent intervals-over multiple cells. This raises the feasibility vs. accuracy dilemma while deciding between different models of mining these stochastic trajectories. We demonstrate that inference based on the Maximum Caliber (MaxCal) principle is the method of choice by critically evaluating its computational efficiency and accuracy against two other typical modeling approaches: (i) a detailed model (DM) with explicit consideration of multiple molecules including protein-promoter interaction, and (ii) a coarse-grain model (CGM) using Hill type functions to model feedback. MaxCal provides a reasonably accurate model while being significantly more computationally efficient than DM and CGM. Furthermore, MaxCal requires minimal assumptions since it is a top-down approach and allows systematic model improvement by including constraints of higher order, in contrast to traditional bottom-up approaches that require more parameters or ad hoc assumptions. Thus, based on efficiency, accuracy, and ability to build minimal models, we propose MaxCal as a superior alternative to traditional approaches (DM, CGM) when inferring underlying details of gene circuits with feedback from limited data.

The Anti-Aggregation Holdase Hsp33 Promotes the Formation of Folded Protein Structures.

Holdase chaperones are known to be central to suppressing aggregation, but how they affect substrate conformations remains poorly understood. Here, we use optical tweezers to study how the holdase Hsp33 alters folding transitions within single maltose binding proteins and aggregation transitions between maltose binding protein substrates. Surprisingly, we find that Hsp33 not only suppresses aggregation but also guides the folding process. Two modes of action underlie these effects. First, Hsp33 binds unfolded chains, which suppresses aggregation between substrates and folding transitions within substrates. Second, Hsp33 binding promotes substrate states in which most of the chain is folded and modifies their structure, possibly by intercalating its intrinsically disordered regions. A statistical ensemble model shows how Hsp33 function results from the competition between these two contrasting effects. Our findings reveal an unexpectedly comprehensive functional repertoire for Hsp33 that may be more prevalent among holdases and dispels the notion of a strict chaperone hierarchy.

An analytical theory to describe sequence-specific inter-residue distance profiles for polyampholytes and intrinsically disordered proteins.

Intrinsically Disordered Proteins (IDPs), unlike folded proteins, lack a unique folded structure and rapidly interconvert among ensembles of disordered states. However, they have specific conformational properties when averaged over their ensembles of disordered states. It is critical to develop a theoretical formalism to predict these ensemble average conformational properties that are encoded in the IDP sequence (the specific order in which amino acids/residues are linked). We present a general heteropolymer theory that analytically computes the ensemble average distance profiles (⟨Rij 2⟩) between any two (i, j) monomers (amino acids for IDPs) as a function of the sequence. Information rich distance profiles provide a detailed description of the IDP in contrast to typical metrics such as scaling exponents, radius of gyration, or end-to-end distance. This generalized formalism supersedes homopolymer-like models or models that are built only on the composition of amino acids but ignore sequence details. The prediction of these distance profiles for highly charged polyampholytes and naturally occurring IDPs unmasks salient features that are hidden in the sequence. Moreover, the model reveals strategies to modulate the entire distance map to achieve local or global swelling/compaction by subtle changes/modifications-such as phosphorylation, a biologically relevant process-in specific hotspots in the sequence. Sequence-specific distance profiles and their modulation have been benchmarked against all-atom simulations. Our new formalism also predicts residue-pair specific coil-globule transitions. The analytical nature of the theory will facilitate design of new sequences to achieve specific target distance profiles with broad applications in synthetic biology and polymer science.

A unified analytical theory of heteropolymers for sequence-specific phase behaviors of polyelectrolytes and polyampholytes.

The physical chemistry of liquid-liquid phase separation (LLPS) of polymer solutions bears directly on the assembly of biologically functional dropletlike bodies from proteins and nucleic acids. These biomolecular condensates include certain extracellular materials and intracellular compartments that are characterized as "membraneless organelles." Analytical theories are a valuable, computationally efficient tool for addressing general principles. LLPS of neutral homopolymers is quite well described by theory, but it has been a challenge to develop general theories for the LLPS of heteropolymers involving charge-charge interactions. Here, we present a theory that combines a random-phase-approximation treatment of polymer density fluctuations and an account of intrachain conformational heterogeneity based on renormalized Kuhn lengths to provide predictions of LLPS properties as a function of pH, salt, and charge patterning along the chain sequence. Advancing beyond more limited analytical approaches, our LLPS theory is applicable to a wide variety of charged sequences ranging from highly charged polyelectrolytes to neutral or nearly neutral polyampholytes. This theory should be useful in high-throughput screening of protein and other sequences for their LLPS propensities and can serve as a basis for more comprehensive theories that incorporate nonelectrostatic interactions. Experimental ramifications of our theory are discussed.

Sequence charge decoration dictates coil-globule transition in intrinsically disordered proteins.

We present an analytical theory to compute conformations of heteropolymers-applicable to describe disordered proteins-as a function of temperature and charge sequence. The theory describes coil-globule transition for a given protein sequence when temperature is varied and has been benchmarked against the all-atom Monte Carlo simulation (using CAMPARI) of intrinsically disordered proteins (IDPs). In addition, the model quantitatively shows how subtle alterations of charge placement in the primary sequence-while maintaining the same charge composition-can lead to significant changes in conformation, even as drastic as a coil (swelled above a purely random coil) to globule (collapsed below a random coil) and vice versa. The theory provides insights on how to control (enhance or suppress) these changes by tuning the temperature (or solution condition) and charge decoration. As an application, we predict the distribution of conformations (at room temperature) of all naturally occurring IDPs in the DisProt database and notice significant size variation even among IDPs with a similar composition of positive and negative charges. Based on this, we provide a new diagram-of-states delineating the sequence-conformation relation for proteins in the DisProt database. Next, we study the effect of post-translational modification, e.g., phosphorylation, on IDP conformations. Modifications as little as two-site phosphorylation can significantly alter the size of an IDP with everything else being constant (temperature, salt concentration, etc.). However, not all possible modification sites have the same effect on protein conformations; there are certain "hot spots" that can cause maximal change in conformation. The location of these "hot spots" in the parent sequence can readily be identified by using a sequence charge decoration metric originally introduced by Sawle and Ghosh. The ability of our model to predict conformations (both expanded and collapsed states) of IDPs at a high-throughput level can provide valuable insights into the different mechanisms by which phosphorylation/charge mutation controls IDP function.

Modulating charge patterning and ionic strength as a strategy to induce conformational changes in intrinsically disordered proteins.

We present an analytical theory to describe conformational changes as a function of salt for polymers with a given sequence of charges. We apply this model to describe Intrinsically Disordered Proteins (IDPs) by explicitly accounting for charged residues and their exact placement in the primary sequence while approximating the effect of non-electrostatic interactions at a mean-field level by effective short-range (two body and three-body) interaction parameters. The effect of ions is introduced by treating electrostatic interactions within Debye-Huckle approximation. Using typical values of the short-range mean-field parameters derived from all-atom Monte Carlo simulations (at zero salt), we predict the conformational changes as a function of salt concentration. We notice that conformational transitions in response to changes in ionic strength strongly depend on sequence specific charge patterning. For example, globule to coil transition can be observed upon increasing salt concentration, in stark contrast to uniformly charged polyelectrolyte theories based on net charge only. In addition, it is possible to observe non-monotonic behavior with salt as well. Drastic differences in salt-induced conformational transitions is also evident between two doubly phosphorylated sequences-derived from the same wild type sequence-that only differ in the site of phosphorylation. Similar effects are also predicted between two sequences derived from the same parent sequence differing by a single site mutation where a negative charge is replaced by a positive charge. These effects are purely a result of charge decoration and can only be understood in terms of metrics based on specific placement of charges, and cannot be explained by models based on charge composition alone. Identifying sequences and hot spots within sequences-for post translational modification or charge mutation-using our high-throughput theory will yield fundamental insights into design and biological regulation mediated by phosphorylation and/or local changes in salt concentration.

Perspective: Maximum caliber is a general variational principle for dynamical systems.

We review here Maximum Caliber (Max Cal), a general variational principle for inferring distributions of paths in dynamical processes and networks. Max Cal is to dynamical trajectories what the principle of maximum entropy is to equilibrium states or stationary populations. In Max Cal, you maximize a path entropy over all possible pathways, subject to dynamical constraints, in order to predict relative path weights. Many well-known relationships of non-equilibrium statistical physics-such as the Green-Kubo fluctuation-dissipation relations, Onsager's reciprocal relations, and Prigogine's minimum entropy production-are limited to near-equilibrium processes. Max Cal is more general. While it can readily derive these results under those limits, Max Cal is also applicable far from equilibrium. We give examples of Max Cal as a method of inference about trajectory distributions from limited data, finding reaction coordinates in bio-molecular simulations, and modeling the complex dynamics of non-thermal systems such as gene regulatory networks or the collective firing of neurons. We also survey its basis in principle and some limitations.

Building Predictive Models of Genetic Circuits Using the Principle of Maximum Caliber.

Learning the underlying details of a gene network is a major challenge in cellular and synthetic biology. We address this challenge by building a chemical kinetic model that utilizes information encoded in the stochastic protein expression trajectories typically measured in experiments. The applicability of the proposed method is demonstrated in an auto-activating genetic circuit, a common motif in natural and synthetic gene networks. Our approach is based on the principle of maximum caliber (MaxCal)-a dynamical analog of the principle of maximum entropy-and builds a minimal model using only three constraints: 1) protein synthesis, 2) protein degradation, and 3) positive feedback. The MaxCal-generated model (described with four parameters) was benchmarked against synthetic data generated using a Gillespie algorithm on a known reaction network (with seven parameters). MaxCal accurately predicts underlying rate parameters of protein synthesis and degradation as well as experimental observables such as protein number and dwell-time distributions. Furthermore, MaxCal yields an effective feedback parameter that can be useful for circuit design. We also extend our methodology and demonstrate how to analyze trajectories that are not in protein numbers but in arbitrary fluorescence units, a more typical condition in experiments. This "top-down" methodology based on minimal information-in contrast to traditional "bottom-up" approaches that require ad hoc knowledge of circuit details-provides a powerful tool to accurately infer underlying details of feedback circuits that are not otherwise visible in experiments and to help guide circuit design.

A theoretical method to compute sequence dependent configurational properties in charged polymers and proteins.

A general formalism to compute configurational properties of proteins and other heteropolymers with an arbitrary sequence of charges and non-uniform excluded volume interaction is presented. A variational approach is utilized to predict average distance between any two monomers in the chain. The presented analytical model, for the first time, explicitly incorporates the role of sequence charge distribution to determine relative sizes between two sequences that vary not only in total charge composition but also in charge decoration (even when charge composition is fixed). Furthermore, the formalism is general enough to allow variation in excluded volume interactions between two monomers. Model predictions are benchmarked against the all-atom Monte Carlo studies of Das and Pappu [Proc. Natl. Acad. Sci. U. S. A. 110, 13392 (2013)] for 30 different synthetic sequences of polyampholytes. These sequences possess an equal number of glutamic acid (E) and lysine (K) residues but differ in the patterning within the sequence. Without any fit parameter, the model captures the strong sequence dependence of the simulated values of the radius of gyration with a correlation coefficient of R(2) = 0.9. The model is then applied to real proteins to compare the unfolded state dimensions of 540 orthologous pairs of thermophilic and mesophilic proteins. The excluded volume parameters are assumed similar under denatured conditions, and only electrostatic effects encoded in the sequence are accounted for. With these assumptions, thermophilic proteins are found-with high statistical significance-to have more compact disordered ensemble compared to their mesophilic counterparts. The method presented here, due to its analytical nature, is capable of making such high throughput analysis of multiple proteins and will have broad applications in proteomic studies as well as in other heteropolymeric systems.

Inherent properties of adenylosuccinate lyase could explain S-Ado/SAICAr ratio due to homozygous R426H and R303C mutations.

Adenylosuccinate lyase (ADSL) is a homotetrameric enzyme involved in the de novo purine biosynthesis pathway and purine nucleotide cycle. Missense mutations in the protein lead to ADSL deficiency, an inborn error of purine metabolism characterized by neurological and physiological symptoms. ADSL deficiency is biochemically diagnosed by elevated levels of succinylaminoimidazolecarboxamide riboside (SAICAr) and succinyladenosine (S-Ado), the dephosphorylated derivatives of the substrates. S-Ado/SAICAr ratios have been associated with three phenotypic groups. Different hypotheses to explain these ratios have been proposed. Recent studies have focused on measuring activity on the substrates independently. However, it is important to examine mixtures of the substrates to determine if mutations affect enzyme activity on both substrates similarly in these conditions. The two substrates may experience an indirect communication due to being acted upon by the same enzyme, altering their activities from the non-competitive case. In this study, we investigate this hidden coupling between the two substrates. We chose two mutations that represent extremes of the phenotype, R426H and R303C. We describe a novel electrochemical-detection method of measuring the kinetic activity of ADSL in solution with its two substrates at varying concentration ratios. Furthermore, we develop an enzyme kinetic model to predict substrate activity from a given ratio of substrate concentrations. Our findings indicate a non-linear dependence of the activities on the substrate ratios due to competitive binding, distinct differences in the behaviors of the different mutations, and S-Ado/SAICAr ratios in patients could be explained by inherent properties of the mutant enzyme.

Physical limits of cells and proteomes.

What are the physical limits to cell behavior? Often, the physical limitations can be dominated by the proteome, the cell's complement of proteins. We combine known protein sizes, stabilities, and rates of folding and diffusion, with the known protein-length distributions P(N) of proteomes (Escherichia coli, yeast, and worm), to formulate distributions and scaling relationships in order to address questions of cell physics. Why do mesophilic cells die around 50 °C? How can the maximal growth-rate temperature (around 37 °C) occur so close to the cell-death temperature? The model shows that the cell's death temperature coincides with a denaturation catastrophe of its proteome. The reason cells can function so well just a few degrees below their death temperature is because proteome denaturation is so cooperative. Why are cells so dense-packed with protein molecules (about 20% by volume)? Cells are packed at a density that maximizes biochemical reaction rates. At lower densities, proteins collide too rarely. At higher densities, proteins diffuse too slowly through the crowded cell. What limits cell sizes and growth rates? Cell growth is limited by rates of protein synthesis, by the folding rates of its slowest proteins, and--for large cells--by the rates of its protein diffusion. Useful insights into cell physics may be obtainable from scaling laws that encapsulate information from protein knowledge bases.

Nonadditive entropies yield probability distributions with biases not warranted by the data.

Different quantities that go by the name of entropy are used in variational principles to infer probability distributions from limited data. Shore and Johnson showed that maximizing the Boltzmann-Gibbs form of the entropy ensures that probability distributions inferred satisfy the multiplication rule of probability for independent events in the absence of data coupling such events. Other types of entropies that violate the Shore and Johnson axioms, including nonadditive entropies such as the Tsallis entropy, violate this basic consistency requirement. Here we use the axiomatic framework of Shore and Johnson to show how such nonadditive entropy functions generate biases in probability distributions that are not warranted by the underlying data.

Competition enhances stochasticity in biochemical reactions.

We study stochastic dynamics of two competing complexation reactions (i) A + B↔AB and (ii) A + C↔AC. Such reactions are common in biology where different reactants compete for common resources--examples range from binding enzyme kinetics to gene expression. On the other hand, stochasticity is inherent in biological systems due to small copy numbers. We investigate the complex interplay between competition and stochasticity, using coupled complexation reactions as the model system. Within the master equation formalism, we compute the exact distribution of the number of complexes to analyze equilibrium fluctuations of several observables. Our study reveals that the presence of competition offered by one reaction (say A + C↔AC) can significantly enhance the fluctuation in the other (A + B↔AB). We provide detailed quantitative estimates of this enhanced fluctuation for different combinations of rate constants and numbers of reactant molecules that are typical in biology. We notice that fluctuations can be significant even when two of the reactant molecules (say B and C) are infinite in number, maintaining a fixed stoichiometry, while the other reactant (A) is finite. This is purely due to the coupling mediated via resource sharing and is in stark contrast to the single reaction scenario, where large numbers of one of the components ensure zero fluctuation. Our detailed analysis further highlights regions where numerical estimates of mass action solutions can differ from the actual averages. These observations indicate that averages can be a poor representation of the system, hence analysis that is purely based on averages such as mass action laws can be potentially misleading in such noisy biological systems. We believe that the exhaustive study presented here will provide qualitative and quantitative insights into the role of noise and its enhancement in the presence of competition that will be relevant in many biological settings.