Principles for Optimal Cooperativity in Allosteric Materials

Physical effects of learning

Interacting many-body physical systems ranging from neural networks in the brain to folding proteins to self-modifying electrical circuits can learn to perform diverse tasks. This learning, both in nature and in engineered systems, can occur through evolutionary selection or through dynamical rules that drive active learning from experience. Here, we show that learning in linear physical networks with weak input signals leaves architectural imprints on the Hessian of a physical system. Compared to a generic organization of the system components, (a) the effective physical dimension of the response to inputs decreases, (b) the response of physical degrees of freedom to random perturbations (or system "susceptibility") increases, and (c) the low-eigenvalue eigenvectors of the Hessian align with the task. Overall, these effects embody the typical scenario for learning processes in physical systems in the weak input regime, suggesting ways of discovering whether a physical network may have been trained.

Learning to learn by using nonequilibrium training protocols for adaptable materials

Evolution in time-varying environments naturally leads to adaptable biological systems that can easily switch functionalities. Advances in the synthesis of environmentally responsive materials therefore open up the possibility of creating a wide range of synthetic materials which can also be trained for adaptability. We consider high-dimensional inverse problems for materials where any particular functionality can be realized by numerous equivalent choices of design parameters. By periodically switching targets in a given design algorithm, we can teach a material to perform incompatible functionalities with minimal changes in design parameters. We exhibit this learning strategy for adaptability in two simulated settings: elastic networks that are designed to switch deformation modes with minimal bond changes and heteropolymers whose folding pathway selections are controlled by a minimal set of monomer affinities. The resulting designs can reveal physical principles, such as nucleation-controlled folding, that enable such adaptability.

Conservation and diversity in allosteric fingerprints of proteins for evolutionary-inspired engineering and design

Hand-in-hand work of physics and evolution delivered protein universe with diversity of forms, sizes, and functions. Pervasiveness and advantageous traits of allostery made it an important component of the protein function regulation, calling for thorough investigation of its structural determinants and evolution. Learning directly from nature, we explored here allosteric communication in several major folds and repeat proteins, including α/β and β-barrels, β-propellers, Ig-like fold, ankyrin and α/β leucine-rich repeat proteins, which provide structural platforms for many different enzymatic and signalling functions. We obtained a picture of conserved allosteric communication characteristic in different fold types, modifications of the structure-driven signalling patterns via sequence-determined divergence to specific functions, as well as emergence and potential diversification of allosteric regulation in multi-domain proteins and oligomeric assemblies. Our observations will be instrumental in facilitating the engineering and de novo design of proteins with allosterically regulated functions, including development of therapeutic biologics. In particular, results described here may guide the identification of the optimal structural platforms (e.g. fold type, size, and oligomerization states) and the types of diversifications/perturbations, such as mutations, effector binding, and order-disorder transition. The tunable allosteric linkage across distant regions can be used as a pivotal component in the design/engineering of modular biological systems beyond the traditional scaffolding function.

A theory of localized excitations in supercooled liquids

A new connection between the structure and dynamics in glass-forming liquids is presented. We show how the origin of spatially localized excitations, as defined by the dynamical facilitation (DF) theory, can be understood from a structure-based framework. This framework is constructed by associating excitation events in the DF theory to hopping events between energy minima in the potential energy landscape (PEL). By reducing the PEL to an equal energy well picture and applying a harmonic approximation, we develop a field theory to describe elastic fluctuations about inherent states, which are energy minimizing configurations of the PEL. We model an excitation as a shear transformation zone (STZ) inducing a localized pure shear deformation onto an inherent state. We connect STZs to T1 transition events that break the elastic bonds holding the local structure of an inherent state. A formula for the excitation energy barrier, denoted as J_σ, is obtained as a function of inherent-state elastic moduli and the radial distribution function. The energy barrier from the current theory is compared to the one predicted by the DF theory where good agreement is found in various two-dimensional continuous poly-disperse atomistic models of glass formers. These results strengthen the role of structure and elasticity in driving glassy dynamics through the creation and relaxation of localized excitations.

Allosteric Type and Pathways Are Governed by the Forces of Protein-Ligand Binding

Allostery is central to many cellular processes, by up- or down-regulating target function. However, what determines the allosteric type remains elusive and currently it is impossible to predict whether the allosteric compounds would activate or inhibit target function before experimental studies. We demonstrated that the allosteric type and allosteric pathways are governed by the forces imposed by ligand binding to target protein using the anisotropic network model and developed an allosteric type prediction method (AlloType). AlloType correctly predicted 13 of the 16 allosteric systems in the data set with experimentally determined protein and complex structures as well as verified allosteric types, which was also used to identify allosteric pathways. When applied to glutathione peroxidase 4, a protein with no complex structure information, AlloType could still be able to predict the allosteric type of the recently reported allosteric activators, demonstrating its potential application in designing specific allosteric drugs and uncovering allosteric mechanisms.

Exploring the Allosteric Territory of Protein Function

While the pervasiveness of allostery in proteins is commonly accepted, we further show the generic nature of allosteric mechanisms by analyzing here transmembrane ion-channel viroporin 3a and RNA-dependent RNA polymerase (RdRp) from SARS-CoV-2 along with metabolic enzymes isocitrate dehydrogenase 1 (IDH1) and fumarate hydratase (FH) implicated in cancers. Using the previously developed structure-based statistical mechanical model of allostery (SBSMMA), we share our experience in analyzing the allosteric signaling, predicting latent allosteric sites, inducing and tuning targeted allosteric response, and exploring the allosteric effects of mutations. This, yet incomplete list of phenomenology, forms a complex and unique allosteric territory of protein function, which should be thoroughly explored. We propose a generic computational framework, which not only allows one to obtain a comprehensive allosteric control over proteins but also provides an opportunity to approach the fragment-based design of allosteric effectors and drug candidates. The advantages of allosteric drugs over traditional orthosteric compounds, complemented by the emerging role of the allosteric effects of mutations in the expansion of the cancer mutational landscape and in the increased mutability of viral proteins, leave no choice besides further extensive studies of allosteric mechanisms and their biomedical implications.

Automatic design of mechanical metamaterial actuators

Mechanical metamaterial actuators achieve pre-determined input–output operations exploiting architectural features encoded within a single 3D printed element, thus removing the need for assembling different structural components. Despite the rapid progress in the field, there is still a need for efficient strategies to optimize metamaterial design for a variety of functions. We present a computational method for the automatic design of mechanical metamaterial actuators that combines a reinforced Monte Carlo method with discrete element simulations. 3D printing of selected mechanical metamaterial actuators shows that the machine-generated structures can reach high efficiency, exceeding human-designed structures. We also show that it is possible to design efficient actuators by training a deep neural network which is then able to predict the efficiency from the image of a structure and to identify its functional regions. The elementary actuators devised here can be combined to produce metamaterial machines of arbitrary complexity for countless engineering applications.

Efficient strategies to optimize metamaterial design for specific applications are urgently needed despite the rapid progress in this area. Here the authors propose a computational method combining an optimization algorithm with discrete element simulations for the automatic design of mechanical metamaterial actuators.

Allosteric communication in molecular machines via information exchange: what can be learned from dynamical modeling

Allosteric regulation is crucial for the operation of protein machines and molecular motors. A major challenge is to characterize and quantify the information exchange underlying allosteric communication between remote functional sites in a protein, and to identify the involved relevant pathways. We review applications of two topical approaches of dynamical protein modeling, a kinetic-based single-molecule stochastic model, which employs information thermodynamics to quantify allosteric interactions, and structure-based coarse-grained modeling to characterize intra-molecular couplings in terms of conformational motions and propagating mechanical strain. Both descriptions resolve the directionality of allosteric responses within a protein, emphasizing the concept of causality as the principal hallmark of protein allostery. We discuss the application of techniques from information thermodynamics to dynamic protein elastic networks and evolutionary designed model structures, and the ramifications for protein allostery.

Direct coupling analysis of epistasis in allosteric materials

In allosteric proteins, the binding of a ligand modifies function at a distant active site. Such allosteric pathways can be used as target for drug design, generating considerable interest in inferring them from sequence alignment data. Currently, different methods lead to conflicting results, in particular on the existence of long-range evolutionary couplings between distant amino-acids mediating allostery. Here we propose a resolution of this conundrum, by studying epistasis and its inference in models where an allosteric material is evolved in silico to perform a mechanical task. We find in our model the four types of epistasis (Synergistic, Sign, Antagonistic, Saturation), which can be both short or long-range and have a simple mechanical interpretation. We perform a Direct Coupling Analysis (DCA) and find that DCA predicts well the cost of point mutations but is a rather poor generative model. Strikingly, it can predict short-range epistasis but fails to capture long-range epistasis, in consistence with empirical findings. We propose that such failure is generic when function requires subparts to work in concert. We illustrate this idea with a simple model, which suggests that other methods may be better suited to capture long-range effects.

Allostery in proteins is the property of highly specific responses to ligand binding at a distant site. To inform protocols of de novo drug design, it is fundamental to understand the impact of mutations on allosteric regulation and whether it can be predicted from evolutionary correlations. In this work we consider allosteric architectures artificially evolved to optimize the cooperativity of binding at allosteric and active site. We first characterize the emergent pattern of epistasis as well as the underlying mechanical phenomena, finding the four types of epistasis (Synergistic, Sign, Antagonistic, Saturation), which can be both short or long-range. The numerical evolution of these allosteric architectures allows us to benchmark Direct Coupling Analysis, a method which relies on co-evolution in sequence data to infer direct evolutionary couplings, in connection to allostery. We show that Direct Coupling Analysis predicts quantitatively point mutation costs but underestimates strong long-range epistasis. We provide an argument, based on a simplified model, illustrating the reasons for this discrepancy. Our analysis suggests neural networks as more promising tool to measure epistasis.

Mechanics of Allostery: Contrasting the Induced Fit and Population Shift Scenarios

In allosteric proteins, binding a ligand can affect function at a distant location, for example, by changing the binding affinity of a substrate at the active site. The induced fit and population shift models, which differ by the assumed number of stable configurations, explain such cooperative binding from a thermodynamic viewpoint. Yet, understanding what mechanical principles constrain these models remains a challenge. Here, we provide an empirical study on 34 proteins supporting the idea that allosteric conformational change generally occurs along a soft elastic mode presenting extended regions of high shear. We argue, based on a detailed analysis of how the energy profile along such a mode depends on binding, that in the induced fit scenario, there is an optimal stiffness ka∗ ∼ 1/N for cooperative binding, where N is the number of residues. We find that the population shift scenario is more robust to mutations affecting stiffness because binding becomes more and more cooperative with stiffness up to the same characteristic value ka∗, beyond which cooperativity saturates instead of decaying. We numerically confirm these findings in a nonlinear mechanical model. Dynamical considerations suggest that a stiffness of order ka∗ is favorable in that scenario as well, supporting that for proper function, proteins must evolve a functional elastic mode that is softer as their size increases. In consistency with this view, we find a fair anticorrelation between the stiffness of the allosteric response and protein size in our data set.

Universality of fold-encoded localized vibrations in enzymes

Enzymes speed up biochemical reactions at the core of life by as much as 15 orders of magnitude. Yet, despite considerable advances, the fine dynamical determinants at the microscopic level of their catalytic proficiency are still elusive. In this work, we use a powerful mathematical approach to show that rate-promoting vibrations in the picosecond range, specifically encoded in the 3D protein structure, are localized vibrations optimally coupled to the chemical reaction coordinates at the active site. Remarkably, our theory also exposes an hithertho unknown deep connection between the unique localization fingerprint and a distinct partition of the 3D fold into independent, foldspanning subdomains that govern long-range communication. The universality of these features is demonstrated on a pool of more than 900 enzyme structures, comprising a total of more than 10,000 experimentally annotated catalytic sites. Our theory provides a unified microscopic rationale for the subtle structure-dynamics-function link in proteins.

Allostery in Its Many Disguises: From Theory to Applications

Allosteric regulation plays an important role in many biological processes, such as signal transduction, transcriptional regulation, and metabolism. Allostery is rooted in the fundamental physical properties of macromolecular systems, but its underlying mechanisms are still poorly understood. A collection of contributions to a recent interdisciplinary CECAM (Center Européen de Calcul Atomique et Moléculaire) workshop is used here to provide an overview of the progress and remaining limitations in the understanding of the mechanistic foundations of allostery gained from computational and experimental analyses of real protein systems and model systems. The main conceptual frameworks instrumental in driving the field are discussed. We illustrate the role of these frameworks in illuminating molecular mechanisms and explaining cellular processes, and describe some of their promising practical applications in engineering molecular sensors and informing drug design efforts.

Revealing evolutionary constraints on proteins through sequence analysis

Statistical analysis of alignments of large numbers of protein sequences has revealed "sectors" of collectively coevolving amino acids in several protein families. Here, we show that selection acting on any functional property of a protein, represented by an additive trait, can give rise to such a sector. As an illustration of a selected trait, we consider the elastic energy of an important conformational change within an elastic network model, and we show that selection acting on this energy leads to correlations among residues. For this concrete example and more generally, we demonstrate that the main signature of functional sectors lies in the small-eigenvalue modes of the covariance matrix of the selected sequences. However, secondary signatures of these functional sectors also exist in the extensively-studied large-eigenvalue modes. Our simple, general model leads us to propose a principled method to identify functional sectors, along with the magnitudes of mutational effects, from sequence data. We further demonstrate the robustness of these functional sectors to various forms of selection, and the robustness of our approach to the identification of multiple selected traits.

Designed Elastic Networks: Models of Complex Protein Machinery

Recently, the design of mechanical networks with protein-inspired responses has become increasingly popular. Here, we review contributions which were motivated by studies of protein dynamics employing coarse-grained elastic network models. First, the concept of evolutionary optimization that we developed to design network structures which execute prescribed tasks is explained. We then review what presumably marks the origin of the idea to design complex functional networks which encode protein-inspired behavior, namely the design of an elastic network structure which emulates the cycles of ATP-powered conformational motion in protein machines. Two recent applications are reviewed. First, the construction of a model molecular motor, whose operation incorporates both the tight coupling power stroke as well as the loose coupling Brownian ratchet mechanism, is discussed. Second, the evolutionary design of network structures which encode optimal long-range communication between remote sites and represent mechanical models of allosteric proteins is presented. We discuss the prospects of designed protein-mimicking elastic networks as model systems to elucidate the design principles and functional signatures underlying the operation of complex protein machinery.

Green function of correlated genes in a minimal mechanical model of protein evolution

The function of proteins arises from cooperative interactions and rearrangements of their amino acids, which exhibit large-scale dynamical modes. Long-range correlations have also been revealed in protein sequences, and this has motivated the search for physical links between the observed genetic and dynamic cooperativity. We outline here a simplified theory of protein, which relates sequence correlations to physical interactions and to the emergence of mechanical function. Our protein is modeled as a strongly coupled amino acid network with interactions and motions that are captured by the mechanical propagator, the Green function. The propagator describes how the gene determines the connectivity of the amino acids and thereby, the transmission of forces. Mutations introduce localized perturbations to the propagator that scatter the force field. The emergence of function is manifested by a topological transition when a band of such perturbations divides the protein into subdomains. We find that epistasis-the interaction among mutations in the gene-is related to the nonlinearity of the Green function, which can be interpreted as a sum over multiple scattering paths. We apply this mechanical framework to simulations of protein evolution and observe long-range epistasis, which facilitates collective functional modes.