Theoretical Studies On The Dynamic Aspects Of Macromolecular Function

National Institute Of Diabetes And Digestive And Kidney Diseases

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Abstract

In this reporting period we published four papers dealing with (1) FRET-based structural biology (2) The influence of enzyme clustering on the rate of cascade reactions (3) The rates of reactions involving diffusive barrier crossing and (4) The transport of cargo along microtubules. They will be briefly described below in the order that they are listed in the bibliography, (1) Single-molecule FRET (smFRET) has become a mainstream technique for studying biomolecular structural dynamics. In these experiments, a molecule is illuminated by a laser, and the donor fluorophore is excited. The donor can emit a photon or transfer the excitation to an acceptor which then can emit a photon of a different color. The rate of transfer depends on the inter-dye distance and this is why there is information about conformational dynamics. The output of these experiments is a sequence of photons with recorded colors and arrival times. The distances between fluorescence labels attached to a molecule fluctuate due to conformational dynamics on a wide range of time scales. The rapid and wide adoption of smFRET experiments by an ever-increasing number of groups has generated significant progress in sample preparation, measurement procedures, data analysis, algorithms and documentation. Several labs that employ smFRET approaches have joined forces to inform the smFRET community about streamlining how to perform experiments and analyze results for obtaining quantitative information on biomolecular structure and dynamics. This position paper describes the current state of the art from different perspectives, points to unresolved methodological issues for quantitative structural studies, provides a set of soft recommendations about which an emerging consensus exists, and lists openly available resources for newcomers and seasoned practitioners. (2) Metabolic enzymes in vivo are not always homogeneously distributed but can form spatial assemblies. The enzymes in such clusters often belong to metabolic pathways where the product of the upstream enzyme (an intermediate) is transferred to the downstream enzyme. The formation of enzyme assemblies facilitates substrate channeling of intermediates, i.e., the intermediate can be transferred to the sequential enzyme without first diffusing to the bulk. Channeling in clusters can be very efficient, even when channeling between a pair of enzymes is improbable. To understand the effect of localization, we develop a theory for cascade reactions converting substrates into intermediates and then into products when the enzymes are localized in clusters. The theory shows that the kinetic scheme that describes the reaction with dispersed enzymes changes as a result of clustering. A new reaction channel, in which the substrate is directly converted into product, appears with a diffusion-influenced rate that is expressed in terms of enzyme catalytic efficiencies, diffusion coefficient, and cluster size. This rate is proportional to the cluster channeling probability, which is the probability that an intermediate is converted into product within the cluster in which the intermediate was formed. Simple analytic formulas allow one to quantify how enzyme clustering can affect product formation and regulate the direction of metabolic reaction flux in biological and synthetic systems. (3) The simplest phenomenological description of an isomerization reaction is based on the kinetic scheme A<=>B that involves forward and backward rate constants, kAB and kBA, for transitions between states A and B of the molecule. The relaxation to equilibrium turns out to be a single exponential, exp-(kAB + kBA)t. The simplest microscopic description of such a reaction is diffusion in the presence of a bistable potential with two basins separated by a high barrier. The problem is to express the phenomenological rate constants in terms of microscopic parameters. This was first done by Kramers who calculated the rate constants for high barriers by setting up a seemingly artificial steady state in which particles are injected on one side of the barrier and removed on the other side. The rate constant is then obtained as the ratio of the nonequilibrium steady-state flux across the barrier to the equilibrium population in the well. An alternative, conceptually more straightforward procedure is to focus on estimating the first nonzero eigenvalue of the Smoluchowski operator that describes the microscopic dynamics. When the barrier separating the basins is high, the inter-basin exchange occurs much more slowly than intra-basin relaxation. This is manifested in the eigenvalue spectrum of the evolution operator. Consequently, the microscopic model also predicts that the relaxation to equilibrium is single exponential, exp -e1t, except at very short times (i.e., on the time scale of the intra-well relaxation). Thus, the rate constant can be obtained by equating the lowest nonzero eigenvalue e1 to the sum of the forward and backward rate constants, kAB + kBA. In reference 3 (which was chosen as Editors Pick) we use variational principle to get an upper bound for the magnitude of the first nonzero eigenvalue. Interestingly, this is analogous to the procedure used in quantum chemistry to get the energy of the first excited state of a molecule. We obtain an elegant expression for the upper bound and show that for high barriers, it reduces to Kramers result, thus placing it on a sound theoretical foundation. Our bound is remarkably accurate. For shallow barrier of 0.5 kcal/mole, it is accurate to better than 5%, even for a modest 2 kcal/mole barrier, it is already better than 1%. (4) In this paper, we developed a unified theory for the effective diffusivity for transport when the drift velocity fluctuates. The motivation was to provide a quantitative description of cargo transport by molecular motors, such as kinesin, along filaments in a cell such as microtubules. Imagine that the motor can switch among conformational states where it moves forward or backwards or stalls. Then at long times the cargo will appear to be diffusing along the microtubule. The question is how to relate the effective diffusion constant to microscopic parameters such as the rates of transition between conformational states that move with different velocities. It is remarkable that this problem of biological interest is just the discrete analog of a classic problem in hydrodynamics called Taylor dispersion. Imagine a particle diffusing in a viscous fluid that is flowing in a cylindrical tube. For laminar flow, the fluid velocity is the highest at the center of the tube. Thus because of radial diffusion, the drift velocity of the particle in the flow direction fluctuates leading to an increase of its diffusion coefficient from its intrinsic value. We were able to treat both discrete and continuous fluctuations in a unified way by showing that the effective diffusivity in both cases is just the time integral of the correlation function of the deviation of the velocity from its average value. We then evaluated this time integral in closed form and showed that our analytic expressions recover a variety of results in the literature obtained in different and complicated ways.