Allosteric conformational ensembles have unlimited capacity for integrating information

The Hill function is the universal Hopfield barrier for sharpness of input-output responses

The Hill functions, [Formula: see text], have been widely used in biology for over a century but, with the exception of [Formula: see text], they have had no justification other than as a convenient fit to empirical data. Here, we show that they are the universal limit for the sharpness of any input-output response arising from a Markov process model at thermodynamic equilibrium. Models may represent arbitrary molecular complexity, with multiple ligands, internal states, conformations, coregulators, etc, under core assumptions that are detailed in the paper. The model output may be any linear combination of steady-state probabilities, with components other than the chosen input ligand held constant. This formulation generalizes most of the responses in the literature. We use a coarse-graining method in the graph-theoretic linear framework to show that two sharpness measures for input-output responses fall within an effectively bounded region of the positive quadrant, [Formula: see text], for any equilibrium model with [Formula: see text] input binding sites. [Formula: see text] exhibits a cusp which approaches, but never exceeds, the sharpness of [Formula: see text], but the region and the cusp can be exceeded when models are taken away from thermodynamic equilibrium. Such fundamental thermodynamic limits are called Hopfield barriers, and our results provide a biophysical justification for the Hill functions as the universal Hopfield barriers for sharpness. Our results also introduce an object, [Formula: see text], whose structure may be of mathematical interest, and suggest the importance of characterizing Hopfield barriers for other forms of cellular information processing.

The Hill function is the universal Hopfield barrier for sharpness of input-output responses

The Hill functions, ℋ h ( x ) = x h / 1 + x h , have been widely used in biology for over a century but, with the exception of ℋ 1 , they have had no justification other than as a convenient fit to empirical data. Here, we show that they are the universal limit for the sharpness of any input-output response arising from a Markov process model at thermodynamic equilibrium. Models may represent arbitrary molecular complexity, with multiple ligands, internal states, conformations, co-regulators, etc, under core assumptions that are detailed in the paper. The model output may be any linear combination of steady-state probabilities, with components other than the chosen input ligand held constant. This formulation generalises most of the responses in the literature. We use a coarse-graining method in the graph-theoretic linear framework to show that two sharpness measures for input-output responses fall within an effectively bounded region of the positive quadrant, Ω m ⊂ ℝ + 2 , for any equilibrium model with m input binding sites. Ω m exhibits a cusp which approaches, but never exceeds, the sharpness of ℋ m but the region and the cusp can be exceeded when models are taken away from thermodynamic equilibrium. Such fundamental thermodynamic limits are called Hopfield barriers and our results provide a biophysical justification for the Hill functions as the universal Hopfield barriers for sharpness. Our results also introduce an object, Ω m , whose structure may be of mathematical interest, and suggest the importance of characterising Hopfield barriers for other forms of cellular information processing.

Cell phenotypes can be predicted from propensities of protein conformations

Proteins exist as dynamic conformational ensembles. Here we suggest that the propensities of the conformations can be predictors of cell function. The conformational states that the molecules preferentially visit can be viewed as phenotypic determinants, and their mutations work by altering the relative propensities, thus the cell phenotype. Our examples include (i) inactive state variants harboring cancer driver mutations that present active state-like conformational features, as in K-Ras4BG12V compared to other K-Ras4BG12X mutations; (ii) mutants of the same protein presenting vastly different phenotypic and clinical profiles: cancer and neurodevelopmental disorders; (iii) alterations in the occupancies of the conformational (sub)states influencing enzyme reactivity. Thus, protein conformational propensities can determine cell fate. They can also suggest the allosteric drugs efficiency.

The linear framework II: using graph theory to analyse the transient regime of Markov processes

The linear framework uses finite, directed graphs with labelled edges to model biomolecular systems. Graph vertices represent chemical species or molecular states, edges represent reactions or transitions and edge labels represent rates that also describe how the system is interacting with its environment. The present paper is a sequel to a recent review of the framework that focussed on how graph-theoretic methods give insight into steady states as rational algebraic functions of the edge labels. Here, we focus on the transient regime for systems that correspond to continuous-time Markov processes. In this case, the graph specifies the infinitesimal generator of the process. We show how the moments of the first-passage time distribution, and related quantities, such as splitting probabilities and conditional first-passage times, can also be expressed as rational algebraic functions of the labels. This capability is timely, as new experimental methods are finally giving access to the transient dynamic regime and revealing the computations and information processing that occur before a steady state is reached. We illustrate the concepts, methods and formulas through examples and show how the results may be used to illuminate previous findings in the literature.

Protein conformational ensembles in function: roles and mechanisms

The sequence-structure-function paradigm has dominated twentieth century molecular biology. The paradigm tacitly stipulated that for each sequence there exists a single, well-organized protein structure. Yet, to sustain cell life, function requires (i) that there be more than a single structure, (ii) that there be switching between the structures, and (iii) that the structures be incompletely organized. These fundamental tenets called for an updated sequence-conformational ensemble-function paradigm. The powerful energy landscape idea, which is the foundation of modernized molecular biology, imported the conformational ensemble framework from physics and chemistry. This framework embraces the recognition that proteins are dynamic and are always interconverting between conformational states with varying energies. The more stable the conformation the more populated it is. The changes in the populations of the states are required for cell life. As an example, in vivo, under physiological conditions, wild type kinases commonly populate their more stable "closed", inactive, conformations. However, there are minor populations of the "open", ligand-free states. Upon their stabilization, e.g., by high affinity interactions or mutations, their ensembles shift to occupy the active states. Here we discuss the role of conformational propensities in function. We provide multiple examples of diverse systems, including protein kinases, lipid kinases, and Ras GTPases, discuss diverse conformational mechanisms, and provide a broad outlook on protein ensembles in the cell. We propose that the number of molecules in the active state (inactive for repressors), determine protein function, and that the dynamic, relative conformational propensities, rather than the rigid structures, are the hallmark of cell life.

Allosterically Driven Assembly of a Multisite Cage-Based [2]Semirotaxane

The assembly of a [2]semirotaxane from a half-dumbbell endowed with a pyrazine coordination site and a bis-Zn(II) porphyrin cage as a multisite ring is reported. The threading is allosterically driven by the coordination of silver(I) ions to the multiple binding sites of the cage linkers, as shown by NMR studies. Addition of chloride ions destabilizes [2]semirotaxane, leading to its disassembly into its cage and half-dumbbell components.

Misguided model of human behavior: Comment on C. H. Burt: "Challenging the utility of polygenic scores for social science…"

This commentary emphasizes two problem areas mentioned by Burt. First, that within-family designs do not eradicate stratification confounds. Second, that the linear/additive model of genetic causes of form and variation is not supported by recent progress in molecular biology. It concludes with an appeal for a (biologically and psychologically) more realistic model of such causes.

Transcriptional kinetic synergy: A complex landscape revealed by integrating modeling and synthetic biology

Transcription factors (TFs) control gene expression, often acting synergistically. Classical thermodynamic models offer a biophysical explanation for synergy based on binding cooperativity and regulated recruitment of RNA polymerase. Because transcription requires polymerase to transition through multiple states, recent work suggests that "kinetic synergy" can arise through TFs acting on distinct steps of the transcription cycle. These types of synergy are not mutually exclusive and are difficult to disentangle conceptually and experimentally. Here, we model and build a synthetic circuit in which TFs bind to a single shared site on DNA, such that TFs cannot synergize by simultaneous binding. We model mRNA production as a function of both TF binding and regulation of the transcription cycle, revealing a complex landscape dependent on TF concentration, DNA binding affinity, and regulatory activity. We use synthetic TFs to confirm that the transcription cycle must be integrated with recruitment for a quantitative understanding of gene regulation.

The structure and function of YTHDF epitranscriptomic m6A readers

Specific RNA sequences modified by a methylated adenosine, N⁶-methyladenosine (m⁶A), contribute to the post-transcriptional regulation of gene expression. The quantity of m⁶A in RNA is orchestrated by enzymes that write and erase it, while its effects are mediated by proteins that bind to read this modification. Dysfunction of this post-transcriptional regulatory process has been linked to human disease. Although the initial focus has been on pharmacological targeting of the writer and eraser enzymes, interest in the reader proteins has been challenged by a lack of clear understanding of their functional roles and molecular mechanisms of action. Readers of m⁶A-modified RNA (m⁶A-RNA) - the YTH (YT521-B homology) domain-containing protein family paralogs 1-3 (YTHDF1-3, referred to here as DF1-DF3) - are emerging as therapeutic targets as their links to pathological processes such as cancer and inflammation and their roles in regulating m⁶A-RNA fate become clear. We provide an updated understanding of the modes of action of DF1-DF3 and review their structures to unlock insights into drug design approaches for DF paralog-selective inhibition.

Exploring and Learning the Universe of Protein Allostery Using Artificial Intelligence Augmented Biophysical and Computational Approaches

Allosteric mechanisms are commonly employed regulatory tools used by proteins to orchestrate complex biochemical processes and control communications in cells. The quantitative understanding and characterization of allosteric molecular events are among major challenges in modern biology and require integration of innovative computational experimental approaches to obtain atomistic-level knowledge of the allosteric states, interactions, and dynamic conformational landscapes. The growing body of computational and experimental studies empowered by emerging artificial intelligence (AI) technologies has opened up new paradigms for exploring and learning the universe of protein allostery from first principles. In this review we analyze recent developments in high-throughput deep mutational scanning of allosteric protein functions; applications and latest adaptations of Alpha-fold structural prediction methods for studies of protein dynamics and allostery; new frontiers in integrating machine learning and enhanced sampling techniques for characterization of allostery; and recent advances in structural biology approaches for studies of allosteric systems. We also highlight recent computational and experimental studies of the SARS-CoV-2 spike (S) proteins revealing an important and often hidden role of allosteric regulation driving functional conformational changes, binding interactions with the host receptor, and mutational escape mechanisms of S proteins which are critical for viral infection. We conclude with a summary and outlook of future directions suggesting that AI-augmented biophysical and computer simulation approaches are beginning to transform studies of protein allostery toward systematic characterization of allosteric landscapes, hidden allosteric states, and mechanisms which may bring about a new revolution in molecular biology and drug discovery.

Competing constraints shape the nonequilibrium limits of cellular decision-making

Gene regulation is central to cellular function. Yet, despite decades of work, we lack quantitative models that can predict how transcriptional control emerges from molecular interactions at the gene locus. Thermodynamic models of transcription, which assume that gene circuits operate at equilibrium, have previously been employed with considerable success in the context of bacterial systems. However, the presence of ATP-dependent processes within the eukaryotic transcriptional cycle suggests that equilibrium models may be insufficient to capture how eukaryotic gene circuits sense and respond to input transcription factor concentrations. Here, we employ simple kinetic models of transcription to investigate how energy dissipation within the transcriptional cycle impacts the rate at which genes transmit information and drive cellular decisions. We find that biologically plausible levels of energy input can lead to significant gains in how rapidly gene loci transmit information but discover that the regulatory mechanisms underlying these gains change depending on the level of interference from noncognate activator binding. When interference is low, information is maximized by harnessing energy to push the sensitivity of the transcriptional response to input transcription factors beyond its equilibrium limits. Conversely, when interference is high, conditions favor genes that harness energy to increase transcriptional specificity by proofreading activator identity. Our analysis further reveals that equilibrium gene regulatory mechanisms break down as transcriptional interference increases, suggesting that energy dissipation may be indispensable in systems where noncognate factor interference is sufficiently large.

Reformulating nonequilibrium steady states and generalized Hopfield discrimination

Despite substantial progress in nonequilibrium physics, steady-state (s.s.) probabilities remain intractable to analysis. For a Markov process, s.s. probabilities can be expressed in terms of transition rates using the graph-theoretic matrix-tree theorem (MTT). The MTT reveals that, away from equilibrium, s.s. probabilities become globally dependent on all rates, with expressions growing exponentially in the system size. This overwhelming complexity and lack of thermodynamic interpretation have greatly impeded analysis. Here we show that s.s. probabilities are proportional to the average of exp(-S(P)), where S(P) is the entropy generated along minimal paths P in the graph, and the average is taken over a probability distribution on spanning trees. Assuming Arrhenius rates, this "arboreal" distribution becomes Boltzmann-like, with the energy of a tree being its total edge barrier energy. This reformulation offers a thermodynamic interpretation that smoothly generalizes equilibrium statistical mechanics and reorganizes the expression complexity: the number of distinct minimal-path entropies depends not on graph size but on the entropy production index, a measure of nonequilibrium complexity. We demonstrate the power of this reformulation by extending Hopfield's analysis of discrimination by kinetic proofreading to any graph with index 1. We derive a general formula for the error ratio and use it to show how local energy dissipation can yield optimal discrimination through global synergy.

Slow dynamics measured by phosphorescence lifetime reveals global conformational changes in human adult hemoglobin induced by allosteric effectors

The mechanism underlying allostery in hemoglobin (Hb) is still not completely understood. Various models describing the action of allosteric effectors on Hb function have been published in the literature. It has also been reported that some allosteric effectors-such as chloride ions, inositol hexaphosphate, 2,3-diphospho-glycerate and bezafibrate-considerably lower the oxygen affinity of Hb. In this context, an important question is the extent to which these changes influence the conformational dynamics of the protein. Earlier, we elaborated a challenging method based on phosphorescence quenching, which makes characterizing protein-internal dynamics possible in the ms time range. The experimental technique involves phosphorescence lifetime measurements in thermal equilibrium at varied temperatures from 10 K up to 273 K, based on the signal of Zn-protoporphyrin substituted for the heme in the β-subunits of Hb. The thermal activation of protein dynamics was observed by the enhancement of phosphorescence quenching attributed to O2 diffusion. It was shown that the thermal activation of protein matrix dynamics was clearly distinguishable from the dynamic activation of the aqueous solvent, and was therefore highly specific for the protein. In the present work, the same method was used to study the changes in the parameters of the dynamic activation of human HbA induced by binding allosteric effectors. We interpreted the phenomenon as phase transition between two states. The fitting of this model to lifetime data yielded the change of energy and entropy in the activation process and the quenching rate in the dynamically activated state. The fitted parameters were particularly sensitive to the presence of allosteric effectors and could be interpreted in line with results from earlier experimental studies. The results suggest that allosteric effectors are tightly coupled to the dynamics of the whole protein, and thus underline the importance of global dynamics in the regulation of Hb function.

The Mediator complex as a master regulator of transcription by RNA polymerase II

The Mediator complex, which in humans is 1.4 MDa in size and includes 26 subunits, controls many aspects of RNA polymerase II (Pol II) function. Apart from its size, a defining feature of Mediator is its intrinsic disorder and conformational flexibility, which contributes to its ability to undergo phase separation and to interact with a myriad of regulatory factors. In this Review, we discuss Mediator structure and function, with emphasis on recent cryogenic electron microscopy data of the 4.0-MDa transcription preinitiation complex. We further discuss how Mediator and sequence-specific DNA-binding transcription factors enable enhancer-dependent regulation of Pol II function at distal gene promoters, through the formation of molecular condensates (or transcription hubs) and chromatin loops. Mediator regulation of Pol II reinitiation is also discussed, in the context of transcription bursting. We propose a working model for Mediator function that combines experimental results and theoretical considerations related to enhancer-promoter interactions, which reconciles contradictory data regarding whether enhancer-promoter communication is direct or indirect. We conclude with a discussion of Mediator's potential as a therapeutic target and of future research directions.

Allostery: Allosteric Cancer Drivers and Innovative Allosteric Drugs

Here, we discuss the principles of allosteric activating mutations, propagation downstream of the signals that they prompt, and allosteric drugs, with examples from the Ras signaling network. We focus on Abl kinase where mutations shift the landscape toward the active, imatinib binding-incompetent conformation, likely resulting in the high affinity ATP outcompeting drug binding. Recent pharmacological innovation extends to allosteric inhibitor (GNF-5)-linked PROTAC, targeting Bcr-Abl1 myristoylation site, and broadly, allosteric heterobifunctional degraders that destroy targets, rather than inhibiting them. Designed chemical linkers in bifunctional degraders can connect the allosteric ligand that binds the target protein and the E3 ubiquitin ligase warhead anchor. The physical properties and favored conformational state of the engineered linker can precisely coordinate the distance and orientation between the target and the recruited E3. Allosteric PROTACs, noncompetitive molecular glues, and bitopic ligands, with covalent links of allosteric ligands and orthosteric warheads, increase the effective local concentration of productively oriented and placed ligands. Through covalent chemical or peptide linkers, allosteric drugs can collaborate with competitive drugs, degrader anchors, or other molecules of choice, driving innovative drug discovery.

Protein Conformational Space at the Edge of Allostery: Turning a Non-allosteric Malate Dehydrogenase into an "Allosterized" Enzyme using Evolution Guided Punctual Mutations

We unveil the intimate relationship between protein dynamics and allostery by following the trajectories of model proteins in their conformational and sequence spaces. Starting from a nonallosteric hyperthermophilic malate dehydrogenase, we have tracked the role of protein dynamics in the evolution of the allosteric capacity. Based on a large phylogenetic analysis of the malate (MalDH) and lactate dehydrogenase (LDH) superfamily, we identified two amino acid positions that could have had a major role for the emergence of allostery in LDHs, which we targeted for investigation by site-directed mutagenesis. Wild-type MalDH and the single and double mutants were tested with respect to their substrate recognition profiles. The double mutant displayed a sigmoid-shaped profile typical of homotropic activation in LDH. By using molecular dynamics simulations, we showed that the mutations induce a drastic change in the protein sampling of its conformational landscape, making transiently T-like (inactive) conformers, typical of allosteric LDHs, accessible. Our data fit well with the seminal key concept linking protein dynamics and evolvability. We showed that the selection of a new phenotype can be achieved by a few key dynamics-enhancing mutations causing the enrichment of low-populated conformational substates.

A New Model of Hemoglobin Oxygenation

The study of hemoglobin oxygenation, starting from the classical works of Hill, has laid the foundation for molecular biophysics. The cooperative nature of oxygen binding to hemoglobin has been variously described in different models. In the Adair model, which better fits the experimental data, the constants of oxygen binding at various stages differ. However, the physical meaning of the parameters in this model remains unclear. In this work, we applied Hill's approach, extending its interpretation; we obtained a good agreement between the theory and the experiment. The equation in which the Hill coefficient is modulated by the Lorentz distribution for oxygen partial pressure approximates the experimental data better than not only the classical Hill equation, but also the Adair equation.

AlphaFold, Artificial Intelligence (AI), and Allostery

AlphaFold has burst into our lives. A powerful algorithm that underscores the strength of biological sequence data and artificial intelligence (AI). AlphaFold has appended projects and research directions. The database it has been creating promises an untold number of applications with vast potential impacts that are still difficult to surmise. AI approaches can revolutionize personalized treatments and usher in better-informed clinical trials. They promise to make giant leaps toward reshaping and revamping drug discovery strategies, selecting and prioritizing combinations of drug targets. Here, we briefly overview AI in structural biology, including in molecular dynamics simulations and prediction of microbiota–human protein–protein interactions. We highlight the advancements accomplished by the deep-learning-powered AlphaFold in protein structure prediction and their powerful impact on the life sciences. At the same time, AlphaFold does not resolve the decades-long protein folding challenge, nor does it identify the folding pathways. The models that AlphaFold provides do not capture conformational mechanisms like frustration and allostery, which are rooted in ensembles, and controlled by their dynamic distributions. Allostery and signaling are properties of populations. AlphaFold also does not generate ensembles of intrinsically disordered proteins and regions, instead describing them by their low structural probabilities. Since AlphaFold generates single ranked structures, rather than conformational ensembles, it cannot elucidate the mechanisms of allosteric activating driver hotspot mutations nor of allosteric drug resistance. However, by capturing key features, deep learning techniques can use the single predicted conformation as the basis for generating a diverse ensemble.

The linear framework: using graph theory to reveal the algebra and thermodynamics of biomolecular systems

The linear framework uses finite, directed graphs with labelled edges to model biomolecular systems. Graph vertices represent biochemical species or molecular states, edges represent reactions or transitions and labels represent rates. The graph yields a linear dynamics for molecular concentrations or state probabilities, with the graph Laplacian as the operator, and the labels encode the nonlinear interactions between system and environment. The labels can be specified by vertices of other graphs or by conservation laws or, when the environment consists of thermodynamic reservoirs, they may be constants. In the latter case, the graphs correspond to infinitesimal generators of Markov processes. The key advantage of the framework has been that steady states are determined as rational algebraic functions of the labels by the Matrix-Tree theorems of graph theory. When the system is at thermodynamic equilibrium, this prescription recovers equilibrium statistical mechanics but it continues to hold for non-equilibrium steady states. The framework goes beyond other graph-based approaches in treating the graph as a mathematical object, for which general theorems can be formulated that accommodate biomolecular complexity. It has been particularly effective at analysing enzyme-catalysed modification systems and input-output responses.

Cellular sentience as the primary source of biological order and evolution

All life is cellular, starting some 4 billion years ago with the emergence of the first cells. In order to survive their early evolution in the face of an extremely challenging environment, the very first cells invented cellular sentience and cognition, allowing them to make relevant decisions to survive through creative adaptations in a continuously running evolutionary narrative. We propose that the success of cellular life has crucially depended on a biological version of Maxwell's demons which permits the extraction of relevant sensory information and energy from the cellular environment, allowing cells to sustain anti-entropic actions. These sensor-effector actions allowed for the creative construction of biological order in the form of diverse organic macromolecules, including crucial polymers such as DNA, RNA, and cytoskeleton. Ordered biopolymers store analogue (structures as templates) and digital (nucleotide sequences of DNA and RNA) information that functioned as a form memory to support the development of organisms and their evolution. Crucially, all cells are formed by the division of previous cells, and their plasma membranes are physically and informationally continuous across evolution since the beginning of cellular life. It is argued that life is supported through life-specific principles which support cellular sentience, distinguishing life from non-life. Biological order, together with cellular cognition and sentience, allow the creative evolution of all living organisms as the authentic authors of evolutionary novelty.

Genes and knowledge: Response to Baverstock, K. the gene an appraisal. https://doi.org/10.1016/j.pbiomolbio.2021.04.005

This response aims to expand on some of the issues raised by Keith Baverstock's The Gene: An Appraisal, especially on the evolution and nature of knowledge in living things. In contrast to the simple associationism envisaged in "genetic information", it emphasises the dynamic complexity and changeability of most natural environments, and, therefore, predictability based on underlying statistical structures. That seems to be the basis of the "cognitive" functions increasingly being reported about cellular, as well as more evolved, functions, and of the autonomous agency of organisms thriving creatively in complex environments.