Steered molecular dynamics (SMD) permits efficient investigations of molecular processes by focusing on selected degrees of freedom. We explain how one can, in the framework of SMD, employ Jarzynski's equality (also known as the nonequilibrium work relation) to calculate potentials of mean force (PMF). We outline the theory that serves this purpose and connects nonequilibrium processes (such as SMD simulations) with equilibrium properties (such as the PMF). We review the derivation of Jarzynski's equality, generalize it to isobaric--isothermal processes, and discuss its implications in relation to the second law of thermodynamics and computer simulations. In the relevant regime of steering by means of stiff springs, we demonstrate that the work on the system is Gaussian-distributed regardless of the speed of the process simulated. In this case, the cumulant expansion of Jarzynski's equality can be safely terminated at second order. We illustrate the PMF calculation method for an exemplary simulation and demonstrate the Gaussian nature of the resulting work distribution.
Jarzynski's equality is applied to free energy calculations from steered molecular dynamics simulations of biomolecules. The helix-coil transition of deca-alanine in vacuum is used as an example. With about ten trajectories sampled, the second order cumulant expansion, among the various averaging schemes examined, yields the most accurate estimates. We compare umbrella sampling and the present method, and find that their efficiencies are comparable.
Aquaglyceroporin GlpF selectively conducts water and linear polyalcohols, such as glycerol, across the inner membrane of Escherichia coli. We report steered molecular dynamics simulations of glycerol conduction through GlpF, in which external forces accelerate the transchannel conduction in a manner that preserves the intrinsic conduction mechanism. The simulations reveal channel-glycerol hydrogen bonding interactions and the stereoselectivity of the channel. Employing Jarzynski's identity between free energy and irreversible work, we reconstruct the potential of mean force along the conduction pathway through a time series analysis of molecular dynamics trajectories. This potential locates binding sites and barriers inside the channel; it also reveals a low energy periplasmic vestibule suited for efficient uptake of glycerol from the environment.A quaporins (1), a family of water transporting membrane proteins, are present in all life forms, and defects in their function cause physiological disorders (2). Among more than 150 members identified to date (2), the Escherichia coli glycerol facilitator (GlpF) belongs to the aquaglyceroporin subclass, which is permeable to both water and glycerol. GlpF also stereoselectively conducts longer linear polyalcohols (3, 4). At physiological conditions, all aquaporins exclude charged solutes, including protons, and thereby preserve the electrochemical potential across the cell membrane (2).A 2.2-Å resolution x-ray structure of GlpF revealed a homotetrameric architecture with glycerol and water present inside the channel (5). Each monomeric channel has two halfmembrane spanning repeats related by a quasi-two-fold symmetry. About half of each repeat is ␣-helical; the other half adopts a particular nonhelical structure (5, 6). The N termini of the helical repeats meet at the Asn-Pro-Ala (NPA) motifs located at the channel center. The NPA motifs are conserved among all aquaporins (2), and their spatial arrangement is critical for the biological function of the channel (6-8). The nonhelical repeats expose the backbone carbonyl groups of residues 64-66 and 195-201 toward the channel interior, where they serve as hydrogen acceptors for the substrate (6). The channel diameter measures less than 3.5 Å at its narrowest point, the selectivity filter (SF), lined with residues Trp-48, 9). In the constriction region of the channel, approximately 25 Å long, the substrate is translocated in a single file following a curvilinear pathway (6), a feature critical for excluding proton conduction (10). The hydroxyl groups of glycerol make hydrogen bonds with exposed carbonyl oxygen atoms, polar hydrogen atoms, and water, whereas the aliphatic backbone of glycerol faces the opposite hydrophobic side of the channel (6). The amphipathic channel lining in GlpF ensures the selectivity for linear polyalcohols (5). The amphipathicity of the channel's interior also seems to be important for water transport and is found, although to a less extent, in pure water-conducting aquaporins, such as aquaporin-1 ...
Most proteins that participate in cellular signalling networks contain modular protein-interaction domains. Multiple versions of such domains are present within a given organism: the yeast proteome, for example, contains 27 different Src homology 3 (SH3) domains. This raises the potential problem of cross-reaction. It is generally thought that isolated domain-ligand pairs lack sufficient information to encode biologically unique interactions, and that specificity is instead encoded by the context in which the interaction pairs are presented. Here we show that an isolated peptide ligand from the yeast protein Pbs2 recognizes its biological partner, the SH3 domain from Sho1, with near-absolute specificity--no other SH3 domain present in the yeast genome cross-reacts with the Pbs2 peptide, in vivo or in vitro. Such high specificity, however, is not observed in a set of non-yeast SH3 domains, and Pbs2 motif variants that cross-react with other SH3 domains confer a fitness defect, indicating that the Pbs2 motif might have been optimized to minimize interaction with competing domains specifically found in yeast. System-wide negative selection is a subtle but powerful evolutionary mechanism to optimize specificity within an interaction network composed of overlapping recognition elements.
How scaffold proteins control information flow in signaling pathways is poorly understood: Do they simply tether components, or do they precisely orient and activate them? We found that the yeast mitogen-activated protein (MAP) kinase scaffold Ste5 is tolerant to major stereochemical perturbations; heterologous protein interactions could functionally replace native kinase recruitment interactions, indicating that simple tethering is largely sufficient for scaffold-mediated signaling. Moreover, by engineering a scaffold that tethers a unique kinase set, we could create a synthetic MAP kinase pathway with non-natural input-output properties. These findings demonstrate that scaffolds are highly flexible organizing factors that can facilitate pathway evolution and engineering.
As most biological species, photosynthetic lifeforms have evolved to function optimally, despite thermal disorder and with fault tolerance. It remains a challenge to understand how this is achieved. To address this challenge the function of the protein-pigment complex photosystem I (PSI) of the cyanobacterium Synechococcus elongatus is investigated theoretically. The recently obtained high resolution structure of this complex exhibits an aggregate of 96 chlorophylls that are electronically coupled to function as a light-harvesting antenna complex. This paper constructs an effective Hamiltonian for the chlorophyll aggregate to describe excitation transfer dynamics and spectral properties of PSI. For this purpose, a new kinetic expansion method, the sojourn expansion, is introduced. Our study shows that at room temperature fluctuations of site energies have little effect on the calculated excitation lifetime and quantum yield, which compare favorably with experimental results. The efficiency of the system is found to be robust against 'pruning' of individual chlorophylls. An optimality of the arrangement of chlorophylls is identified through the quantum yield in comparison with an ensemble of randomly oriented chlorophylls, though, the quantum yield is seen to change only within a narrow interval in such an ensemble.
Purple bacteria have developed an efficient apparatus to harvest sunlight. The apparatus consists of up to four types of pigment−protein complexes: (i) the photosynthetic reaction center surrounded by (ii) the light-harvesting complex LH1, (iii) antenna complexes LH2, which are replaced under low-light conditions by (iv) antenna complexes LH3 with a higher absorption maximum. Following absorption of light anywhere in the apparatus, electronic excitation is transferred between the pigment−protein complexes until it is used for the primary photoreaction in the reaction center. We calculate, using Förster theory, all rates for the inter-complex excitation transfer processes on the basis of the atomic level structures of the pigment−protein complexes and of an effective Hamiltonian, established previously, for intracomplex excitations. The kinetics of excitation migration in the photosynthetic apparatus is described through a master equation which connects the calculated transfer rates to the overall architecture of the apparatus. For two exemplary architectures the efficiency, distribution of dissipation, and time evolution of excitation migration are determined. Pigment−protein complexes are found to form an excitation reservoir, in which excitation is spread over many chromophores rather than forming an excitation funnel in which excitation is transferred without detours from the periphery to the RC. This feature permits a high quantum yield of 83% to 89%, but also protects the apparatus from overheating by spreading dissipation over all complexes. Substitution of LH2 complexes by LH3 complexes or changing an architecture in which few LH2 (LH3) complexes are in contact with LH1 to an architecture in which all LH2 (LH3) complexes are in contact with LH1 increases the quantum yield up to 94% and decreases the degree to which dissipation is evenly distributed.
Long timescale (>1 micros) molecular dynamics simulations of protein folding offer a powerful tool for understanding the atomic-scale interactions that determine a protein's folding pathway and stabilize its native state. Unfortunately, when the simulated protein fails to fold, it is often unclear whether the failure is due to a deficiency in the underlying force fields or simply a lack of sufficient simulation time. We examine one such case, the human Pin1 WW domain, using the recently developed deactivated morphing method to calculate free energy differences between misfolded and folded states. We find that the force field we used favors the misfolded states, explaining the failure of the folding simulations. Possible further applications of deactivated morphing and implications for force field development are discussed.
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