Physical Biology

Nanobodies, like other antibodies bind their targets through complementarity determining regions (CDRs). Improving nanobody-antigen binding affinities by introducing mutations in these CDRs is critical for biotechnological applications. However, any mutation is expected to introduce changes in the behavior of the protein, such as fluctuations of residues, correlation of fluctuations of residue pairs, response of a residue to perturbation of another. Most importantly, the nanoscale dynamics of the protein may change. In these respects, the problem is similar to the general problem of dynamic allostery, a perturbation at one site affecting the response at another site. Using the Gaussian Network Model of proteins, we show that CDR mutations indeed modify the fluctuation profile and dynamics of the nanobody. Effects are not confined to CDR regions but extend throughout the full structure. We introduce a computational scheme where fluctuations of a residue are perturbed by a force and response amplitude and response time of the remaining residues are determined. Response to a perturbation of a residue shows a synchronous and an asynchronous component. The model is used to quantify the effects of mutation on protein dynamics: highly perturbable residues and highly responsive residues of the nanobody are determined. Residues whose perturbation has no effect on protein behavior may also be determined with the present model. Three known nanobodies produced by nature are used as an illustrative example and their various modifications which we generated by CDR residue mutations are analyzed.

1Structural changes in proteins allow them to exist in several conformations. Non-covalent interactions with ligands drive the structural changes, thereby allowing the protein to perform its biological function. Recent findings suggest that many proteins are always in an equilibrium of different conformations and that each of these conformations can be formed by both the ligand-free and ligand-bound protein. By using classical statistical mechanics, we derived the equilibrium probabilities of forming a conformation with and without ligand. We found, under certain conditions, that increasing the probability of forming a conformation by the ligand-free protein also increases the probability of forming the same conformation when the protein has a ligand bound. Further, we found that changes in the conformational equilibrium of the ligand-free protein can increase or decrease the affinity for the ligand.

Over the past three decades, the senior author had interacted with and been mentored by the late Professor Robert M. Nerem. In his memory, the authors summarized several observations made, ideas conceptualized, and mathematical models developed during this period for quantitatively analyzing memory effects in repetitive protein-protein interactions (PPI). Interactions between proteins in an organism coordinate its biological processes and may impact its responses to changing environment and diseases through feedback systems. Feedback systems function by using changes in the past to influence behaviors in the future, which we refer here as memory. Specifically, we consider how proteins on cell or in isolation retain information about prior interactions to impact current interactions. The micropipette, biomembrane force probe and atomic force microscopic techniques were used to repeatedly assay several PPIs. The resulting time series were analyzed by a previous and two new models to extract three memory indices of short (seconds), intermediate (minutes), and long (hours) timescales. We found that interactions of cell membrane, but not soluble, T cell receptor (TCR) with peptide-major histocompatibility complex (pMHC) exhibits short-term memory that impacts on-rate, but not off-rate of the binding kinetics. Peptide dissociation from MHC resulted in intermediate- and long-term memories in TCR-pMHC interactions. However, we observed no changes in kinetic parameters by repetitive measurements on living cells over intermediate timescale using stable pMHCs. Parameters quantifying memory effects in PPIs could provide additional information regarding biological mechanisms. The methods developed herein also provide tools for future research.

Time resolved Raman and infrared spectroscopy experiments show the basic features of information transfer between residues in proteins. Here, we present the theoretical basis of information transfer using a simple elastic net model and recently developed entropy transfer concept in proteins. Mutual information between two residues is a measure of communication in proteins which shows the maximum amount of information that may be transferred between two residues. However, it does not explain the actual amount of transfer nor the transfer rate of information between residues. For this, dynamic equations of the system are needed. We used the Schreiber theory of information transfer and the Gaussian network Model of proteins, together with the solution of the Langevin equation, to quantify allosteric information transfer. Results of the model are in perfect agreement with ultraviolet resonance Raman measurements. Analysis of the allosteric protein Human NAD-dependent isocitrate dehydrogenase shows that a multitude of paths contribute collectively to information transfer. While the peak values of information transferred are small relative to information content of residues, considering the estimated transfer rates, which are in the order of megabits per second, sustained transfer during the activity time-span of proteins may be significant.

Agent based models were used to describe electrical signalling in bacterial biofilms in three dimensions. Specifically, wavefronts of potassium ions in E. coli biofilms subjected to stress from blue light were modelled from experimental data. Electrical signalling only occurs when the biofilms grow beyond a threshold size, which we have shown to vary with the K+ ion diffusivity and the K+ ion threshold concentration which triggered firing in the fire-diffuse-fire model. The transport of the propagating wavefronts shows super-diffusive scaling on time. K+ ion diffusivity is the main factor that affects the wavefront velocity. The K+ ion diffusivity and the firing threshold also affect the anomalous exponent for the propagation of the wavefront determining whether the wavefront is sub-diffusive or super-diffusive. The geometry of the biofilm and its relation to the mean square displacement (MSD) of the wavefront as a function of time was investigated for spherical, cylindrical, cubical and mushroom-like structures. The MSD varied significantly with geometry; an additional regime to the kinetics occurred when the potassium wavefront leaves the biofilm. Adding cylindrical defects to the biofilm, which are known to occur in E. coli biofilms, the wavefront MSD also had an extra kinetic regime for the propagation through the defect.

Stochastic resonance (SR) describes the synchronization between noise of a system and an applied oscillating field to achieve an optimized response signal. In this work, we use simulations to investigate the phenomenon of SR of a single stranded DNA driven through a nanopore when an oscillating electric field is added. The system is comprised of a MspA protein nanopore embedded in a membrane and different lengths of DNA is driven from one end of the pore to the other via a constant potential difference. We superimposed an oscillating electric field on top of the existing electric field. The source of noise is due to thermal fluctuations, since the system is immersed in solution at room temperature. Here, the signal optimization we seek is the increase in translocation time of DNA through the protein nanopore. Normally, translocation time scales linearly with DNA length and inversely with driving force in a drift dominated regime. We found a non-monotonic dependence of the mean translocation time with the frequency of the oscillating field. This non-monotonic behavior of the translocation time is observed for all lengths of DNA, but SR occurs only for longer DNA. Furthermore, we also see evidence of DNA extension being influenced by the oscillating field while moving through the nanopore.

A prototypical morphogen gradient that plays a key role in the early embryonic development of fruit flies, by providing positional information to cells, is that of the transcription factor Bicoid (Bcd). Recently a one-dimensional quantum walk model has been utilised to explain its multiple dynamic modes observed through fluorescence correlation spectroscopy (FCS) studies using a closed quantum system approach. In this work we use an open quantum system approach to the dynamics of the Bcd gradient formation and show that exactly the same dynamics are obtained through this more rigorous analysis. We then use the thus obtained expression for the fast dynamic modes to explain the Bcd transcription factor search times for binding to the promoter regions along the DNA. Specifically, we find that the large values of diffusivity allowed by quantum mechanics can avoid the paradox of faster-than-diffusion association rates without any need for the transcription factor to constantly alternate between 1D and 3D diffusion-based search processes. This might help explain the fast and precise transcriptional response elicited by such factors. We conclude that, since many transcription factors share a common search strategy for target gene regulatory regions, our mechanism may have a wide range of applicability.

Biological systems frequently contain biochemical species present as small numbers of slowly diffusing molecules, leading to fluctuations that invalidate deterministic analyses of system dynamics. The development of mathematical tools that account for the spatial distribution and discrete number of reacting molecules is vital for understanding cellular behavior and engineering biological circuits. Here we present an algorithm for an event-driven stochastic spatiotemporal simulation of a general reaction process that bridges well-mixed and unmixed systems. The algorithm is based on time-varying particle probability density functions whose overlap in time and space is proportional to reactive propensity. We show this to be mathematically equivalent to the Gillespie algorithm in the specific case of fast diffusion. We develop a computational implementation of this algorithm and provide a Fourier transformation-based approach which allows for near constant computational complexity with respect to the number of individual particles of a given species. To test this simulation method, we examine reaction and diffusion limited regimes of a bimolecular association-dissociation reaction. In the reaction limited regime where mixing occurs between individual reactions, equilibrium numbers of components match the expected values from mean field methods. In the diffusion limited regime, however, spatial correlations between newly dissociated species persist, leading to rebinding events and a shift the in the observed molecular counts. In the final part of this work, we examine how changes in enzyme efficiency can emerge from changes in diffusive mobility alone, as may result from protein complex formation.

In many fields there is interest in manipulating genes and gene networks to realize improved trait phenotypes. The practicality of doing so, however, requires accepted theory on the properties of gene networks that is well-tested by empirical results. The extension of quantitative genetics to include models that incorporate properties of gene networks expands the long tradition of studying epistasis resulting from gene-gene interactions. Here we consider NK models of gene networks by applying concepts from graph theory and Boolean logic theory, motivated by a desire to model the parameters that influence predictive skill for trait phenotypes under the control of gene networks; N defines the number of graph nodes, the number of genes in the network, and K defines the number of edges per node in the graph, representing the gene-gene interactions. We define and consider the attractor period of an NK network as an emergent trait phenotype for our purposes. A long-standing theoretical treatment of the dynamical properties of random Boolean networks suggests a transition from long to short attractor periods as a function of the average node degree K and the bias probability P in the applied Boolean rules. In this paper we investigate the appropriateness of this theory for predicting trait phenotypes on random and real microorganism networks through numerical simulation. We show that: (i) the transition zone between long and short attractor periods depends on the number of network nodes for random networks; (ii) networks derived from metabolic reaction data on microorganisms also show a transition from long to short attractor periods, but at higher values of the bias probability than in random networks with similar numbers of network nodes and average node degree; (iii) the distribution of phenotypes measured on microorganism networks shows more variation than random networks when the bias probability in the Boolean rules is above 0.75; and (iv) the topological structure of networks built from metabolic reaction data is not random, being best approximated, in a statistical sense, by a lognormal distribution. The implications of these results for predicting trait phenotypes where the genetic architecture of a trait is a gene network are discussed.

Experiments and theories have shown that when steric interactions between crowding particles and proteins are dominant, which give rise to Asakura-Oosawa depletion forces, then the stabilities of the proteins increase compared to the infinite dilution case. We show using theoretical arguments that the crowder volume fraction ({Phi}C) dependent increase in the melting temperature of globular proteins, [Formula], where [Formula]. The effective Flory exponent,{nu} eff, relates the radius of gyration in the unfolded state to the number of amino acid residues in the protein. We derive the bound 1.25 [≤] [≤] 2.0. The theoretical predictions are confirmed using molecular simulations of {lambda} repressor in the presence of spherical crowding particles. Analyses of previous simulations and experiments confirm the predicted theoretical bound for . We show that the non-specific attractions between crowding particles and amino acid residues have to be substantial to fully negate the enhanced protein stabilities due to intra protein attractive Asakura-Oosawa (AO) depletion potential. Using the findings, we provide an alternate explanation for the very modest (often less than 0.5 Kcal/mol) destabilization in certain proteins in the cellular milieu. Cellular environment is polydisperse containing large and small crowding agents. AO arguments suggest that proteins would be localized between large (sizes exceeding that of the proteins) crowders, which are predicted to have negligible effect on stability. In vitro experiments containing mixtures of crowding particles could validate or invalidate the predictions.

Traditional information theoretic analysis of functionally conserved binding interactions described by multiple sequence alignments are unable to provide direct insights into the underlying strength, spatial distribution, and coordination of the biophysical motions that govern protein binding interactions during signaling and regulatory function. However, molecular dynamic (MD) simulations of proteins in bound vs. unbound conformational states can allow for the combined application of machine learning classification and information theory towards many problems posed by comparative protein dynamics. After both bound and unbound protein dynamic states are adequately sampled in MD software, they can be employed as a comparative training set for a binary classifier capable of discerning the complex dynamical consequences of protein binding interactions with DNA or other proteins. The statistical validation of the learner on MD simulations of homologs can be used to assess its ability to recognize functional protein motions that are conserved over evolutionary time scales. Regions of proteins with functionally conserved dynamics are identifiable by their ability to induce significant correlations in local learning performance across homologous MD simulations. Through case studies of Rbp subunit 4/7 interaction in RNA Pol II and DNA-protein interactions of TATA binding protein, we demonstrate this method of detecting functionally conserved protein dynamics. We also demonstrate how the concepts of relative entropy (i.e. information gain) and mutual information applied to the binary classification states of MD simulations can be used to compare the impacts of molecular variation on conserved dynamics and to identify coordinated motions involved in dynamic interactions across sites.

Genes are connected in complex networks of interactions where often the product of one gene is a transcription factor that alters the expression of another. Many of these networks are based on a few fundamental motifs leading to switches and oscillators of various kinds. And yet, there is more to the story than which transcription factors control these various circuits. These transcription factors are often themselves under the control of effector molecules that bind them and alter their level of activity. Traditionally, much beautiful work has shown how to think about the stability of the different states achieved by these fundamental regulatory architectures by examining how parameters such as transcription rates, degradation rates and dissociation constants tune the circuit, giving rise to behavior such as bistability. However, such studies explore dynamics without asking how these quantities are altered in real time in living cells as opposed to at the fingertips of the synthetic biologists pipette or on the computational biologists computer screen. In this paper, we make a departure from the conventional dynamical systems view of these regulatory motifs by using statistical mechanical models to focus on endogenous signaling knobs such as effector concentrations rather than on the convenient but more experimentally remote knobs such as dissociation constants, transcription rates and degradation rates that are often considered. We also contrast the traditional use of Hill functions to describe transcription factor binding with more detailed thermodynamic models. This approach provides insights into how biological parameters are tuned to control the stability of regulatory motifs in living cells, sometimes revealing quite a different picture than is found by using Hill functions and tuning circuit parameters by hand.

Due to the high concentration of proteins, nucleic acids and other macromolecules, the bacterial cytoplasm is typically described as a crowded environment. However, the extent to which each of these macromolecules individually affects the mobility of macromolecular complexes, and how this depends on growth conditions, is presently unclear. In this study, we sought to quantify the crowding experienced by an exogenous 40 nm fluorescent particle in the cytoplasm of E. coli under different growth conditions. By performing single particle tracking measurements in cells selectively depleted of DNA and/or mRNA, we determined the contribution to crowding of mRNA, DNA and remaining cellular components, i.e., mostly proteins and ribosomes. To estimate this contribution to crowding, we quantified the difference of the particles diffusion coefficient in conditions with and without those macromolecules. We found that the contributions of the three classes of components were of comparable magnitude, being largest in the case of proteins and ribosomes. We further found that the contributions of mRNA and DNA to crowding were significantly larger than expected based on their volumetric fractions alone. Finally, we found that the crowding contributions change only slightly with the growth conditions. These results reveal how various cellular components partake in crowding of the cytoplasm and the consequences this has for the mobility of large macromolecular complexes. Statement of SignificanceThe mobility of a particle of interest in the cytoplasm depends on a variety of factors that include the concentration, shape and physicochemical properties of crowding obstacles. Different macromolecules in the cell are therefore expected to hinder the mobility of a given particle to different extents. However, an accurate and systematic investigation of these hindrances to mobility in vivo has not been yet carried out. In this work, through a novel combination of experimental and computational approaches, we determine the diffusion coefficient of a 40 nm particle in the cytoplasm of E. coli under conditions of selective removal of some macromolecules. This allows us to quantify the hindering effect of each of the depleted macromolecules on the mobility of the said particle. For DNA, mRNA, and remaining macromolecules, we observe that this effect is of comparable magnitude, being largest in the latter case. This work sheds light on the interplay between intracellular composition and the physical properties of the cytoplasm at the 40 nm scale.

Biological organisms have sensors that communicate information about the environment. Analyzing how well these biological sensors function has usually been done with mutual information between the sensor signal and the environment, but that can be computationally intractable and summarize something quite complex with just a single number. We suggest that alternatively, one may profitably analyze these biosensors using bias and variance or confusion matrices, depending on the kind of environment. Stimulus-dependent Maximum Entropy models are used to develop estimators of the environmental state given the sensor state, and these estimators in turn are then used to calculate either the bias and variance of the estimator or confusion matrices. We focus on several examples to understand the utility of non-information-based analyses: ligand-receptor binding models spanning genetic regulation to neuronal communication to bacterial chemotaxis, and spin-glass Ising models for neural activity in cultured neurons. These new computationally-efficient analyses add insight to existing analyses based on mutual information; in particular, mutual information estimates give one number to characterize responses to all environmental inputs, and this analysis method characterizes how sensors respond to each environmental input. Categorical analyses, meanwhile, indicate the presence of memory without much prediction in confusion matrix elements in cultured neural networks, adding to previous understanding from mutual information estimates. Author summaryAll living organisms use external stimuli to navigate their environment via their sensors. Because encoding information costs energy, organisms retain only a fraction of the information received from their sensors, ideally information that maximizes their ability to remember past environmental states or predict future ones, key functions that support survival. To better understand how well sensor systems absorb stimulus information, we used stimulus-dependent Maximum Entropy (MaxEnt) models with maximum likelihood estimation and typical statistical metrics, such as confusion matrices or bias and variance. This approach provides two primary benefits over previous approaches: it is more computationally efficient, and it provides a more information-rich picture on how sensors interact with stimulus.

Ligand-receptor systems, covalent modification cycles, and transcriptional networks are the fundamental components of cell signaling and gene expression systems. While their behavior in reaching a steady state regime under step-like stimulation is well understood, their response under repetitive stimulation, particularly at early time stages is poorly characterized. This is despite the fact that early-stage responses to external inputs are arguably as informative as late-stage ones. In simple systems, a periodic stimulation elicits an initial transient response, followed by periodic behavior. Transient responses are relevant when the stimulation has a limited time span, or when the stimulated components timescale is slow as compared to the timescales of the downstream processes, in which case these fast processes may be capturing only those transients. In this study, we analyze the frequency response of simple motifs at different time stages. We use dose-conserved pulsatile input signals, meaning that the amplitude or the duration of the pulses varies along with frequency to conserve input dose, and consider different metrics versus frequency curves. We show that in ligand-receptor systems, there is a frequency preference response (band-pass filter) in some specific metrics during the transient stages, which is not present in the periodic regime. We suggest this is a general system-level mechanism that cells may use to filter input signals that have consequences for higher order circuits. Additionally, we evaluate how the described behavior in isolated motifs is reflected in similar types of responses in cascades and pathways of which they are a part. Our studies suggest that transient frequency preferences are important dynamic features of cell signaling and gene expression systems, which have been overlooked.

Cooperative response is ubiquitous and vital for regulatory control and ultra-sensitivity in various cellular biophysical processes. Ligands, acting as signaling molecules, carry information which is transmitted through the elements of the biochemical network during binding processes. In this work, we address a fundamental issue regarding the link between the information content of the various states of the binding network and the observable binding statistics. Two seminal models of cooperativity, viz., the Koshland-Nemethy-Filmer (KNF) network and the Monod-Wyman-Changeux (MWC) network are considered for this purpsoe which are solved using the chemical master equation approach. Our results establish that the variation of Shannon information associated with the network states has a generic form related to the average binding number. Further, the logarithmic sensitivity of the slope of Shannon information is shown to be related to the Hill slope in terms of the variance of the binding number distributions. 1

FRET-sensors are a well-established method to investigate protein-protein interactions. To determine how FRET-sensor can be employed for the study of switchable allosteric modulator proteins (SAMPs) I extend a previously established model for enzymatic SAMPs to include a FRET-sensor system. Using this model, I determine the prerequisites for using FRET to investigate modulator-regulator interaction. The model shows, that under saturating stimulus conditions only a trimolecular complex contributes to the measured FRET value. How the signal is relayed by the modulator can be investigated by comparing FRET values of unstimulated and signal-saturated sensor systems. Finally, to determine the allosteric mode of signal transduction the natural logarithm of the ratio of stimulated and unstimulated FRET efficiencies is a useful metric.

Actin is an intracellular protein that constitutes a primary component of the cellular cytoskeleton and is accordingly crucial for various cell functions. Actin assembles into semi-flexible filaments that cross-link to form higher order structures within the cytoskeleton. In turn, the actin cytoskeketon regulates cell shape, and participates in cell migration and division. A variety of theoretical models have been proposed to investigate actin dynamics across distinct scales, from the stochastic nature of protein and molecular motor dynamics to the deterministic macroscopic behavior of the cytoskeleton. Yet, the relationship between molecular-level actin processes and cellular-level actin network behavior remains understudied, where prior models do not holistically bridge the two scales together. In this work, we focus on the dynamics of the formation of a branched actin structure as observed at the leading edge of motile eukaryotic cells. We construct a minimal agent-based model for the microscale branching actin dynamics, and a deterministic partial differential equation model for the macroscopic network growth and bulk diffusion. The microscale model is stochastic, as its dynamics are based on molecular level effects. The effective diffusion constant and reaction rates of the deterministic model are calculated from averaged simulations of the microscale model, using the mean displacement of the network front and characteristics of the actin network density. With this method, we design concrete metrics that connect phenomenological parameters in the reaction-diffusion system to the biochemical molecular rates typically measured experimentally. A parameter sensitivity analysis in the stochastic agent-based model shows that the effective diffusion and growth constants vary with branching parameters in a complementary way to ensure that the outward speed of the network remains fixed. These results suggest that perturbations to microscale rates can have significant consequences at the macroscopic level, and these should be taken into account when proposing continuum models of actin network dynamics.

AO_SCPLOWBSTRACTC_SCPLOWAllostery in proteins is a phenomenon in which the binding of a ligand induces alterations in the activity of remote functional sites. This can be conceptually viewed as point-to-point telecommunication in a networked communication medium, where a signal (ligand) arriving at the input (binding site) propagates through the network (interconnected and interacting atoms) to reach the output (remote functional site). The reliable transmission of the signal to distal points occurs despite all the disturbances (noise) affecting the protein. Based on this point of view, we propose a computational frequency-domain framework to characterize the displacements and the fluctuations in a region within the protein, originating from the ligand excitation at the binding site and noise, respectively. We characterize the displacements in the presence of the ligand, and the fluctuations in its absence. In the former case, the effect of the ligand is modeled as an external dynamic oscillatory force excitation, whereas in the latter, the sole source of fluctuations is the noise arising from the interactions with the surrounding medium that is further shaped by the internal protein network dynamics. We introduce the excitation frequency as a key factor in a Signal-to-Noise ratio (SNR) based analysis, where SNR is defined as the ratio of the displacements stemming from only the ligand to the fluctuations due to noise alone. We then employ an information-theoretic (communication) channel capacity analysis that extends the SNR based characterization by providing a route for discovering new allosteric regions. We demonstrate the potential utility of the proposed methods for the representative PDZ3 protein.

Nanoparticles have the potential to enhance therapeutic success and reduce toxicity-based treatment side effects via the targeted delivery of drugs to cells. This delivery relies on complex interactions between numerous biological, chemical and physical processes. The intertwined nature of these processes has thus far hindered attempts to understand their individual impact. Variation in experimental data, such as the number of nanoparticles inside each cell, further inhibits understanding. Here we present a mathematical framework that is capable of examining the impact of individual processes during nanoparticle delivery. We demonstrate that variation in experimental nanoparticle uptake data can be explained by three factors: random nanoparticle motion; variation in nanoparticle-cell interactions; and variation in the maximum nanoparticle uptake per cell. Without all three factors, the experimental data cannot be explained. This work provides insight into biological mecha-nisms that cause heterogeneous responses to treatment, and enables precise identification of treatment-resistant cell subpopulations.