Landscape-Based View on the Stepping Movement of Myosin VI.

Myosin VI dimer walks toward the minus end of the actin filament with a large and variable step size of 25-36 nm. Two competing models have been put forward to explain this large step size. The Spudich model assumes that the myosin VI dimer associates at a distal tail near the cargo-binding domain, which makes two full-length single α-helix (SAH) domains serve as long legs. In contrast, the Houdusse-Sweeney model assumes that the association occurs in the middle (between residues 913 and 940) of the SAH domain and that the three-helix bundles unfold to ensure the large step size. Their consistency with the observation of stepping motion with a large and variable step size has not been examined in detail. To compare the two proposed models of myosin VI, we computationally characterized the free energy landscape experienced by the leading head during the stepping movement along the actin filament using the elastic network model of two heads and an implicit model of the SAH domains. Our results showed that the Spudich model is more consistent with the 25-36 nm step size than the Houdusse-Sweeney model. The unfolding of the three-helix bundles gives rise to the free energy bias toward a shorter distance between two heads. Besides, the stiffness of the SAH domain is a key factor for giving strong energetic bias toward the longer distance of stepping. Free energy analysis of the stepping motion complements the visual inspection of static structures and enables a deeper understanding of underlying mechanisms of molecular motors.

Single-molecular and ensemble-level oscillations of cyanobacterial circadian clock.

When three cyanobacterial proteins, KaiA, KaiB, and KaiC, are incubated with ATP in vitro, the phosphorylation level of KaiC hexamers shows stable oscillation with approximately 24 h period. In order to understand this KaiABC clockwork, we need to analyze both the macroscopic synchronization of a large number of KaiC hexamers and the microscopic reactions and structural changes in individual KaiC molecules. In the present paper, we explain two coarse-grained theoretical models, the many-molecule (MM) model and the single-molecule (SM) model, to bridge the gap between macroscopic and microscopic understandings. In the simulation results with these models, ATP hydrolysis in the CI domain of KaiC hexamers drives oscillation of individual KaiC hexamers and the ATP hydrolysis is necessary for synchronizing oscillations of a large number of KaiC hexamers. Sensitive temperature dependence of the lifetime of the ADP bound state in the CI domain makes the oscillation period temperature insensitive. ATPase activity is correlated to the frequency of phosphorylation oscillation in the single molecule of KaiC hexamer, which should be the origin of the observed ensemble-level correlation between the ATPase activity and the frequency of phosphorylation oscillation. Thus, the simulation results with the MM and SM models suggest that ATP hydrolysis stochastically occurring in each CI domain of individual KaiC hexamers is a key process for oscillatory behaviors of the ensemble of many KaiC hexamers.

Role of ATP Hydrolysis in Cyanobacterial Circadian Oscillator.

A cyanobacterial protein KaiC shows a stable oscillation in its phosphorylation level with approximately one day period when three proteins, KaiA, KaiB, and KaiC, are incubated in the presence of ATP in vitro. During this oscillation, KaiC hydrolyzes more ATP molecules than required for phosphorylation. Here, in this report, a theoretical model of the KaiABC oscillator is developed to elucidate the role of this ATP consumption by assuming multifold feedback relations among reactions and structural transition in each KaiC molecule and the structure-dependent binding reactions among Kai proteins. Results of numerical simulation showed that ATP hydrolysis is a driving mechanism of the phosphorylation oscillation in the present model, and that the frequency of ATP hydrolysis in individual KaiC molecules is correlated to the frequency of oscillation in the ensemble of many Kai molecules, which indicates that the coherent oscillation is generated through the coupled microscopic intramolecular and ensemble-level many-molecular regulations.

Importance of consensus region of multiple-ligand templates in a virtual screening method.

We discuss methods and ideas of virtual screening (VS) for drug discovery by examining the performance of VS-APPLE, a recently developed VS method, which extensively utilizes the tendency of single binding pockets to bind diversely different ligands, i.e. promiscuity of binding pockets. In VS-APPLE, multiple ligands bound to a pocket are spatially arranged by maximizing structural overlap of the protein while keeping their relative position and orientation with respect to the pocket surface, which are then combined into a multiple-ligand template for screening test compounds. To greatly reduce the computational cost, comparison of test compound structures are made only with limited regions of the multiple-ligand template. Even when we use the narrow regions with most densely populated atoms for the comparison, VSAPPLE outperforms other conventional VS methods in terms of Area Under the Curve (AUC) measure. This region with densely populated atoms corresponds to the consensus region among multiple ligands. It is typically observed that expansion of the sampled region including more atoms improves screening efficiency. However, for some target proteins, considering only a small consensus region is enough for the effective screening of test compounds. These results suggest that the performance test of VS methods sheds light on the mechanisms of protein-ligand interactions, and elucidation of the protein-ligand interactions should further help improvement of VS methods.

Cooperativity and modularity in protein folding.

A simple statistical mechanical model proposed by Wako and Saitô has explained the aspects of protein folding surprisingly well. This model was systematically applied to multiple proteins by Muñoz and Eaton and has since been referred to as the Wako-Saitô-Muñoz-Eaton (WSME) model. The success of the WSME model in explaining the folding of many proteins has verified the hypothesis that the folding is dominated by native interactions, which makes the energy landscape globally biased toward native conformation. Using the WSME and other related models, Saitô emphasized the importance of the hierarchical pathway in protein folding; folding starts with the creation of contiguous segments having a native-like configuration and proceeds as growth and coalescence of these segments. The Φ-values calculated for barnase with the WSME model suggested that segments contributing to the folding nucleus are similar to the structural modules defined by the pattern of native atomic contacts. The WSME model was extended to explain folding of multi-domain proteins having a complex topology, which opened the way to comprehensively understanding the folding process of multi-domain proteins. The WSME model was also extended to describe allosteric transitions, indicating that the allosteric structural movement does not occur as a deterministic sequential change between two conformations but as a stochastic diffusive motion over the dynamically changing energy landscape. Statistical mechanical viewpoint on folding, as highlighted by the WSME model, has been renovated in the context of modern methods and ideas, and will continue to provide insights on equilibrium and dynamical features of proteins.

VS-APPLE: A Virtual Screening Algorithm Using Promiscuous Protein-Ligand Complexes.

As the number of structurally resolved protein-ligand complexes increases, the ligand-binding pockets of many proteins have been found to accommodate multiple different compounds. Effective use of these structural data is important for developing virtual screening (VS) methods that identify bioactive compounds. Here, we introduce a VS method, VS-APPLE (Virtual Screening Algorithm using Promiscuous Protein-Ligand complExes), based on promiscuous protein-ligand binding structures. In VS-APPLE, multiple ligands bound to a pocket are combined into a query template for screening. Both the structural match between a test compound and the multiple-ligand template and the possible collisions between the test compound and the target protein are evaluated by an efficient geometric hashing method. The performance of VS-APPLE was examined on a filtered, clustered version of the Directory of Useful Decoys data set. In Area Under the Curve analyses of this data set, VS-APPLE outperformed several popular screening programs. Judging from the performance of VS-APPLE, the structural data of promiscuous protein-ligand bindings could be further analyzed and exploited for developing VS methods.

Folding pathway of a multidomain protein depends on its topology of domain connectivity.

How do the folding mechanisms of multidomain proteins depend on protein topology? We addressed this question by developing an Ising-like structure-based model and applying it for the analysis of free-energy landscapes and folding kinetics of an example protein, Escherichia coli dihydrofolate reductase (DHFR). DHFR has two domains, one comprising discontinuous N- and C-terminal parts and the other comprising a continuous middle part of the chain. The simulated folding pathway of DHFR is a sequential process during which the continuous domain folds first, followed by the discontinuous domain, thereby avoiding the rapid decrease in conformation entropy caused by the association of the N- and C-terminal parts during the early phase of folding. Our simulated results consistently explain the observed experimental data on folding kinetics and predict an off-pathway structural fluctuation at equilibrium. For a circular permutant for which the topological complexity of wild-type DHFR is resolved, the balance between energy and entropy is modulated, resulting in the coexistence of the two folding pathways. This coexistence of pathways should account for the experimentally observed complex folding behavior of the circular permutant.

Coupling of lever arm swing and biased Brownian motion in actomyosin.

An important unresolved problem associated with actomyosin motors is the role of Brownian motion in the process of force generation. On the basis of structural observations of myosins and actins, the widely held lever-arm hypothesis has been proposed, in which proteins are assumed to show sequential structural changes among observed and hypothesized structures to exert mechanical force. An alternative hypothesis, the Brownian motion hypothesis, has been supported by single-molecule experiments and emphasizes more on the roles of fluctuating protein movement. In this study, we address the long-standing controversy between the lever-arm hypothesis and the Brownian motion hypothesis through in silico observations of an actomyosin system. We study a system composed of myosin II and actin filament by calculating free-energy landscapes of actin-myosin interactions using the molecular dynamics method and by simulating transitions among dynamically changing free-energy landscapes using the Monte Carlo method. The results obtained by this combined multi-scale calculation show that myosin with inorganic phosphate (Pi) and ADP weakly binds to actin and that after releasing Pi and ADP, myosin moves along the actin filament toward the strong-binding site by exhibiting the biased Brownian motion, a behavior consistent with the observed single-molecular behavior of myosin. Conformational flexibility of loops at the actin-interface of myosin and the N-terminus of actin subunit is necessary for the distinct bias in the Brownian motion. Both the 5.5-11 nm displacement due to the biased Brownian motion and the 3-5 nm displacement due to lever-arm swing contribute to the net displacement of myosin. The calculated results further suggest that the recovery stroke of the lever arm plays an important role in enhancing the displacement of myosin through multiple cycles of ATP hydrolysis, suggesting a unified movement mechanism for various members of the myosin family.

Conformational flexibility of loops of myosin enhances the global bias in the actin-myosin interaction landscape.

A long-standing controversy on the mechanism of an actomyosin motor is the role of the Brownian motion of the myosin head in force generation. In order to shed light on this problem, we calculate free-energy landscapes of interaction between an actin filament and the head (S1) of myosin II by using a coarse-grained model of actomyosin. The results show that the free-energy landscape has a global gradient toward the strong-binding site on actin filament, which explains the biased Brownian motion of myosin S1 observed in a single-molecule experiment [Kitamura et al., Nature, 1999, 397, 129 and Biophysics, 2005, 1, 1]. The distinct global gradient in the landscape is brought about only when the conformation of loop 2 at the actin interface of myosin S1 is flexible. The conformational flexibility of loop 3 also contributes to the gradient in the landscape by compensating the role of loop 2. Though the structure of loop 2 is expanded in the weak-binding state, loop 2 shows the larger fluctuation of compaction and expansion due to the actin-myosin interactions as myosin S1 moves toward the strong-binding site on actin filament. Hence, the increase in the compaction-expansion fluctuation of loop 2, the stronger binding of myosin to actin, and the biased Brownian motion of myosin S1 are coupled with each other and should take place in a concurrent way. This predicted coupling should provide opportunities to further test the hypothesis of the biased Brownian motion in actomyosin.

Time scales in epigenetic dynamics and phenotypic heterogeneity of embryonic stem cells.

A remarkable feature of the self-renewing population of embryonic stem cells (ESCs) is their phenotypic heterogeneity: Nanog and other marker proteins of ESCs show large cell-to-cell variation in their expression level, which should significantly influence the differentiation process of individual cells. The molecular mechanism and biological implication of this heterogeneity, however, still remain elusive. We address this problem by constructing a model of the core gene-network of mouse ESCs. The model takes account of processes of binding/unbinding of transcription factors, formation/dissolution of transcription apparatus, and modification of histone code at each locus of genes in the network. These processes are hierarchically interrelated to each other forming the dynamical feedback loops. By simulating stochastic dynamics of this model, we show that the phenotypic heterogeneity of ESCs can be explained when the chromatin at the Nanog locus undergoes the large scale reorganization in formation/dissolution of transcription apparatus, which should have the timescale similar to the cell cycle period. With this slow transcriptional switching of Nanog, the simulated ESCs fluctuate among multiple transient states, which can trigger the differentiation into the lineage-specific cell states. From the simulated transitions among cell states, the epigenetic landscape underlying transitions is calculated. The slow Nanog switching gives rise to the wide basin of ESC states in the landscape. The bimodal Nanog distribution arising from the kinetic flow running through this ESC basin prevents transdifferentiation and promotes the definite decision of the cell fate. These results show that the distribution of timescales of the regulatory processes is decisively important to characterize the fluctuation of cells and their differentiation process. The analyses through the epigenetic landscape and the kinetic flow on the landscape should provide a guideline to engineer cell differentiation.

Entropic mechanism of allosteric communication in conformational transitions of dihydrofolate reductase.

The mechanism of allosteric conformational transitions of Escherichia coli dihydrofolate reductase (DHFR) is investigated theoretically by applying a newly developed coarse-grained model. Functional forms of interaction potentials in the model depend on the local structural environments around those interactions to represent the many-residue effects due to atomic packing in each local region, and hence, this model is called "the chameleon model". The chameleon model consistently describes the free-energy landscape of two conformational transitions in the catalytic cycle of DHFR, which we call conformational transition 1 (CT1) and conformational transition 2 (CT2); CT1 is accompanied by the hydride transfer reaction, and CT2 is accompanied by the product ligand release. The transition state of CT1 is entropically stabilized by the disordering of loops at the peripheral regions of the protein, which enhances the positively correlated fluctuations at the center part of the protein, showing that the allosteric communication between distant regions through the central region is intrinsically associated with the entropic stabilization of the transition state. The transition state of CT2 is entropically stabilized through the mechanism that enhances the breathing motion of two domains, showing that the difference in the distribution of interactions brings about the difference in the transition mechanism between CT1 and CT2. The chameleon model opens a way to consistently describe the dynamical energy landscape of enzymatic reactions.

Dynamical modeling of three-dimensional genome organization in interphase budding yeast.

Eukaryotic genome is organized in a set of chromosomes each of which consists of a chain of DNA and associated proteins. Processes involving DNA such as transcription, duplication, and repair, therefore, should be intrinsically related to the three-dimensional organization of the genome. In this article, we develop a computational model of the three-dimensional organization of the haploid genome of interphase budding yeast by regarding chromosomes as chains moving under the constraints of nuclear structure and chromatin-chromatin interactions. The simulated genome structure largely fluctuates with the diffusive movement of chromosomes. This fluctuation, however, is not completely random, as parts of chromosomes distribute in characteristic ways to form "territories" in the nucleus. By suitably taking account of constraints arising from the data of the chromosome-conformation-capture measurement, the model explains the observed fluorescence data of chromosome distributions and motions.

Synchronization of circadian oscillation of phosphorylation level of KaiC in vitro.

In recent experimental reports, robust circadian oscillation of the phosphorylation level of KaiC has been reconstituted by incubating three cyanobacterial proteins, KaiA, KaiB, and KaiC, with ATP in vitro. This reconstitution indicates that protein-protein interactions and the associated ATP hydrolysis suffice to generate the oscillation, and suggests that the rhythm arising from this protein-based system is the circadian clock pacemaker in cyanobacteria. The mechanism of this reconstituted oscillation, however, remains elusive. In this study, we extend our previous model of oscillation by explicitly taking two phosphorylation sites of KaiC into account and we apply the extended model to the problem of synchrony of two oscillatory samples mixed at different phases. The agreement between the simulated and observed data suggests that the combined mechanism of the allosteric transition of KaiC hexamers and the monomer shuffling between them plays a key role in synchronization among KaiC hexamers and hence underlies the population-level oscillation of the ensemble of Kai proteins. The predicted synchronization patterns in mixtures of unequal amounts of two samples provide further opportunities to experimentally check the validity of the proposed mechanism. This mechanism of synchronization should be important in vivo for the persistent oscillation when Kai proteins are synthesized at random timing in cyanobacterial cells.

Unidirectional Brownian motion observed in an in silico single molecule experiment of an actomyosin motor.

The actomyosin molecular motor, the motor composed of myosin II and actin filament, is responsible for muscle contraction, converting chemical energy into mechanical work. Although recent single molecule and structural studies have shed new light on the energy-converting mechanism, the physical basis of the molecular-level mechanism remains unclear because of the experimental limitations. To provide a clue to resolve the controversy between the lever-arm mechanism and the Brownian ratchet-like mechanism, we here report an in silico single molecule experiment of an actomyosin motor. When we placed myosin on an actin filament and allowed myosin to move along the filament, we found that myosin exhibits a unidirectional Brownian motion along the filament. This unidirectionality was found to arise from the combination of a nonequilibrium condition realized by coupling to the ATP hydrolysis and a ratchet-like energy landscape inherent in the actin-myosin interaction along the filament, indicating that a Brownian ratchet-like mechanism contributes substantially to the energy conversion of this molecular motor.

Mechanism of robust circadian oscillation of KaiC phosphorylation in vitro.

By incubating the mixture of three cyanobacterial proteins, KaiA, KaiB, and KaiC, with ATP in vitro, T. Kondo and his colleagues in recent work reconstituted the robust circadian rhythm of the phosphorylation level of KaiC. This finding indicates that protein-protein interactions and the associated hydrolysis of ATP suffice to generate the circadian rhythm. Several theoretical models have been proposed to explain the rhythm generated in this "protein-only" system, but the clear criterion to discern different possible mechanisms was not known. In this article, we discuss a model based on two basic assumptions: the assumption of the allosteric transition of a KaiC hexamer and the assumption of the monomer exchange between KaiC hexamers. The model shows a stable rhythmic oscillation of the phosphorylation level of KaiC, which is robust against changes in concentration of Kai proteins. We show that this robustness gives a clue to distinguish different possible mechanisms. We also discuss the robustness of oscillation against the change in the system size. Behaviors of the system with the cellular or subcellular size should shed light on the role of the protein-protein interactions in in vivo circadian oscillation.

Monomer-shuffling and allosteric transition in KaiC circadian oscillation.

Circadian rhythms in living organisms have long been attributed solely to a transcription-translation loop comprising a negative or positive feedback. The rhythms in cyanobacteria are known to be modulated by kaiC, kaiA and kaiB genes. It was recently shown, however, that their product proteins KaiC, KaiA and KaiB are sufficient to reconstitute the circadian rhythm in the phosphorylation level of KaiC in vitro. It has since been unclear why such an oscillatory behavior can occur in the absence of the apparent transcription-translation feedback. In the meantime, it has been reported that the monomer exchange between KaiC hexamers occurs in a phosphorylation-dependent manner, which suggests that the monomer shuffling is also involved in the circadian rhythm (H. Kageyama et al., Mol. Cell, 23, 161 (2006)). To further clarify the role of the monomer shuffling, we have performed a computational modeling of interactions among Kai proteins assuming the allosteric transition of KaiC hexamer as well as the monomer shuffling. The results show that the existence of both monomer shuffling and allosteric transition can synchronize the phosphorylation level of the KaiC hexamers, and stabilizes its oscillation.

Correlation between evolutionary structural development and protein folding.

Evolution should have played important roles in determining folding mechanisms and structures of proteins. In this article we discuss how the folding mechanisms had been affected by the early stage of evolution through which the uniqueness of structure had developed. Although the process of such early-time evolution has remained a mystery, a plausible scenario is that the evolution of proteins toward the ordered structures was guided by functional selection pressure as demonstrated in vitro and in silico. We examine the in silico functional selection of sequences and show that there is a significant correlation between two different processes toward the unique 3D structure, the evolutionary development of structure through sequence selection, and the folding process of the resultant sequence. This finding could be rephrased as protein folding recapitulates the emergence of topology in the molecular evolution. The correlation suggests a guideline for engineering foldable proteins.

Conformational change of the actomyosin complex drives the multiple stepping movement.

Actin-myosin (actomyosin) generates mechanical force by consuming ATP molecules. We apply the energy landscape perspective to address a controversial issue as to whether the myosin head moves with multiple steps after a single ATP hydrolysis or only a single mechanical event of the lever-arm swinging follows a single ATP hydrolysis. Here we propose a theoretical model in which the refolding of the partially unfolded actomyosin complex and the movement of the myosin head along the actin filament are coupled. A single ATP hydrolysis is followed by the formation of a high free-energy partially unfolded actomyosin complex, which then gradually refolds with a concomitant multiple stepping movement on the way to the lowest free-energy rigor state. The model quantitatively explains the single-molecular observation of the multiple stepping movement and is consistent with structural observations of the disorder in the actomyosin-binding process. The model also explains the observed variety in dwell time before each step, which is not accounted for by previous models, such as the lever-arm or ratchet models.