Detailed investigations of the electronic structure and bonding scenario in different carbene–phosphinidenes have been presented using state-of-the-art computational methods (BP86/def2-TZVPP//BP86/def2-SVP). We have endeavored to find the correlation of the calculated 31P chemical shifts with different bonding parameters of compounds to access the relative π-acceptor strengths of the carbenes. 31P chemical shifts exhibit a weak correlation with σ-polarizations of Ccarb–P bonds toward phosphorus; however excellent correlations are obtained in the case of π-polarizations of Ccarb–P bonds toward the carbene carbon (Ccarb) and NPA charges on phosphorus atoms. 31P chemical shifts also show excellent correlations with the electron densities and energy densities of Ccarb–P bonds at BCPs, as suggested by QTAIM calculations. Moreover, EDA-NOCV analysis is implemented to gain brief insight into the bonding scenario in this class of compounds. Good correlation exists between the interaction energies between the carbene and PPh fragments and 31P chemical shifts. Additionally, we have investigated the correlations of calculated 31P chemical shifts with different bonding parameters of the corresponding free carbenes. The bonding scenario in different carbene-substituted phosphinidenes is also explored to see how the bonding situation depends on various substituents on phosphinidenes. The other substituted carbene–phosphinidenes show correlations similar to those of carbene–phenylphosphinidenes.
Signal transmission at the molecular level in many biological complexes occurs through allosteric transitions. Allostery describes the responses of a complex to binding of ligands at sites that are spatially well separated from the binding region. We describe the structural perturbation method, based on phonon propagation in solids, which can be used to determine the signal-transmitting allostery wiring diagram (AWD) in large but finite-sized biological complexes. Application to the bacterial chaperonin GroEL–GroES complex shows that the AWD determined from structures also drives the allosteric transitions dynamically. From both a structural and dynamical perspective these transitions are largely determined by formation and rupture of salt-bridges. The molecular description of allostery in GroEL provides insights into its function, which is quantitatively described by the iterative annealing mechanism. Remarkably, in this complex molecular machine, a deep connection is established between the structures, reaction cycle during which GroEL undergoes a sequence of allosteric transitions, and function, in a self-consistent manner.
This article is part of a discussion meeting issue ‘Allostery and molecular machines’.
Myosin VI (MVI) is the only known member of the myosin superfamily that, upon dimerization, walks processively toward the pointed end of the actin filament. The leading head of the dimer directs the trailing head forward with a power stroke, a conformational change of the motor domain exaggerated by the lever arm. Using a unique coarse-grained model for the power stroke of a single MVI, we provide the molecular basis for its motility. We show that the power stroke occurs in two major steps. First, the motor domain attains the poststroke conformation without directing the lever arm forward; and second, the lever arm reaches the poststroke orientation by undergoing a rotational diffusion. From the analysis of the trajectories, we discover that the potential that directs the rotating lever arm toward the poststroke conformation is almost flat, implying that the lever arm rotation is mostly uncoupled from the motor domain. Because a backward load comparable to the largest interhead tension in a MVI dimer prevents the rotation of the lever arm, our model suggests that the leading-head lever arm of a MVI dimer is uncoupled, in accord with the inference drawn from polarized total internal reflection fluorescence (polTIRF) experiments. Without any adjustable parameter, our simulations lead to quantitative agreement with polTIRF experiments, which validates the structural insights. Finally, in addition to making testable predictions, we also discuss the implications of our model in explaining the broad step-size distribution of the MVI stepping pattern.
Signaling in enzymatic networks is typically triggered by environmental fluctuations, resulting in a series of stochastic chemical reactions, leading to corruption of the signal by noise. For example, information flow is initiated by binding of extracellular ligands to receptors, which is transmitted through a cascade involving kinase-phosphatase stochastic chemical reactions. For a class of such networks, we develop a general field-theoretic approach to calculate the error in signal transmission as a function of an appropriate control variable. Application of the theory to a simple push-pull network, a module in the kinase-phosphatase cascade, recovers the exact results for error in signal transmission previously obtained using umbral calculus [Hinczewski and Thirumalai, Phys. Rev. X 4, 041017 (2014)]. We illustrate the generality of the theory by studying the minimal errors in noise reduction in a reaction cascade with two connected push-pull modules. Such a cascade behaves as an effective three-species network with a pseudointermediate. In this case, optimal information transfer, resulting in the smallest square of the error between the input and output, occurs with a time delay, which is given by the inverse of the decay rate of the pseudointermediate. Surprisingly, in these examples the minimum error computed using simulations that take nonlinearities and discrete nature of molecules into account coincides with the predictions of a linear theory. In contrast, there are substantial deviations between simulations and predictions of the linear theory in error in signal propagation in an enzymatic push-pull network for a certain range of parameters. Inclusion of second-order perturbative corrections shows that differences between simulations and theoretical predictions are minimized. Our study establishes that a field theoretic formulation of stochastic biological signaling offers a systematic way to understand error propagation in networks of arbitrary complexity.
Folded states of single domain globular proteins are compact with high packing density. The radius of gyration, Rg, of both the folded and unfolded states increase as Nν where N is the number of amino acids in the protein. The values of the Flory exponent ν are, respectively, ≈⅓ and ≈0.6 in the folded and unfolded states, coinciding with those for homopolymers. However, the extent of compaction of the unfolded state of a protein under low denaturant concentration (collapsibility), conditions favoring the formation of the folded state, is unknown. We develop a theory that uses the contact map of proteins as input to quantitatively assess collapsibility of proteins. Although collapsibility is universal, the propensity to be compact depends on the protein architecture. Application of the theory to over two thousand proteins shows that collapsibility depends not only on N but also on the contact map reflecting the native structure. A major prediction of the theory is that β-sheet proteins are far more collapsible than structures dominated by α-helices. The theory and the accompanying simulations, validating the theoretical predictions, provide insights into the differing conclusions reached using different experimental probes assessing the extent of compaction of proteins. By calculating the criterion for collapsibility as a function of protein length we provide quantitative insights into the reasons why single domain proteins are small and the physical reasons for the origin of multi-domain proteins. Collapsibility of non-coding RNA molecules is similar β-sheet proteins structures adding support to “Compactness Selection Hypothesis”.
Fluctuations in the physical properties of biological machines are inextricably linked to their functions. Distributions of run lengths and velocities of processive molecular motors, like kinesin-1, are accessible through single-molecule techniques, but rigorous theoretical models for these probabilities are lacking. Here, we derive exact analytic results for a kinetic model to predict the resistive force (F)-dependent velocity [P(v)] and run length [P(n)] distribution functions of generic finitely processive molecular motors. Our theory quantitatively explains the zero force kinesin-1 data for both P(n) and P(v) using the detachment rate as the only parameter. In addition, we predict the F dependence of these quantities. At nonzero F, P(v) is non-Gaussian and is bimodal with peaks at positive and negative values of v, which is due to the discrete step size of kinesin-1. Although the predictions are based on analyses of kinesin-1 data, our results are general and should hold for any processive motor, which walks on a track by taking discrete steps.
One of the most intriguing results of single-molecule experiments on proteins and nucleic acids is the discovery of functional heterogeneity: the observation that complex cellular machines exhibit multiple, biologically active conformations. The structural differences between these conformations may be subtle, but each distinct state can be remarkably long-lived, with interconversions between states occurring only at macroscopic timescales, fractions of a second or longer. Although we now have proof of functional heterogeneity in a handful of systems-enzymes, motors, adhesion complexes-identifying and measuring it remains a formidable challenge. Here, we show that evidence of this phenomenon is more widespread than previously known, encoded in data collected from some of the most well-established single-molecule techniques: atomic force microscopy or optical tweezer pulling experiments. We present a theoretical procedure for analyzing distributions of rupture/unfolding forces recorded at different pulling speeds. This results in a single parameter, quantifying the degree of heterogeneity, and also leads to bounds on the equilibration and conformational interconversion timescales. Surveying 10 published datasets, we find heterogeneity in 5 of them, all with interconversion rates slower than 10 s(-1) Moreover, we identify two systems where additional data at realizable pulling velocities is likely to find a theoretically predicted, but so far unobserved crossover regime between heterogeneous and nonheterogeneous behavior. The significance of this regime is that it will allow far more precise estimates of the slow conformational switching times, one of the least understood aspects of functional heterogeneity.