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, our research efforts have resulted in five publications in the following areas: a) single-molecule fluorescence spectroscopy 1-2; b) kinetics of diffusion-influenced multisite phosphorylation 3; c) chemical reaction rate theory 4-5. Below, we discuss our findings and contributions in each of these areas. In single-molecule free diffusion experiments, molecules predominantly reside outside a laser spot, occasionally emitting bursts of photons as they pass through the focal spot. These bursts contain valuable information and therefore must be selected using physically reasonable criteria. The accurate analysis of the selected bursts must consider the precise way they were chosen. We propose novel methods to determine the brightness and diffusivity of individual molecule species from the recorded photon arrival times in bursts 1. We derive analytical expressions for the distribution of inter-photon times (with and without burst selection), the distribution of the number of photons in a burst, and the distribution of photons in a burst with recorded arrival times. The theory accurately treats the bias introduced due to the burst selection criteria. We use a Maximum Likelihood (ML) method to find the molecules photon count rate and diffusion coefficient from three kinds of data, i.e., the bursts of photons with recorded arrival times (burstML), inter-photon times in bursts (iptML), and the number of photon counts in a burst (pcML). The new methods were tested using simulated photon trajectories and experimental data from a fluorophore, Atto 488. The new methods provide valuable tools for investigations of complex experimental systems. In collaboration with Dr. Hoi Sung Chung from LCP and his group, we applied the above methods to address the challenging task of characterizing oligomers formed during amyloid 42 (A42) aggregation 2. These oligomers are believed to be toxic species associated with various diseases, but their heterogeneous nature and transient appearance make them difficult to study using traditional methods. To overcome these limitations, we employed a newly developed method based on molecular diffusion theory and maximum likelihood approach 1. We demonstrate that the photon count rate, diffusion time, population, and FRET efficiency can be accurately determined from simulated data and the experimental data of a known oligomerization system, the tetramerization domain of p53. Applying this method to characterize the oligomers of A42, we found that the average size of these oligomers is 70-mer and their overall population is remarkably low, less than 1 nM, during the early and middle stages of aggregation of 1 M A42 peptide. Furthermore, based on their average size and long diffusion time, we predict that the oligomers have a highly elongated rod-like shape 2. Another important contribution made during this reporting period is the development of a new theory of how the diffusive motion of the reactants influences the time dependence of concentrations in multisite phosphorylation 3. It is now well established that post-translational modification by phosphorylation is an important mechanism to regulate and alter protein function. Multisite phosphorylation of a protein can lead to unltrasensitivity, switch-like behavior between bistable steady states, and oscillatory behavior. In order to react, the reactants have to diffuse together and reorient their active sites. The simplest way to account for this effect is to replace the chemical rate constants by the diffusion-influenced ones. However, when two or more sites are modified by an enzyme, this naive theory deviates significantly from the results of particle-based simulations. This is due to pseudo-processivity that arises because of diffusion. Specifically, after phosphorylating the first site, the enzyme and product may not completely diffuse apart. Thus the same enzyme molecule can phosphorylate the second site of the substrate before diffusing away. We have previously shown that new transitions between the reactants must also be introduced. Here we extend our results by considering enzymes that are inactive after modifying the substrate and need time to become active again. This generalization leads to a surprising result. The introduction of enzyme reactivation results in a diffusion-modified kinetic scheme with a new transition that has a negative rate constant. The reason for this is that mapping non-Markovian rate equations onto Markovian ones with time-independent rate constants is not a good approximation at short times. We then developed a non-Markovian theory that involves memory kernels instead of rate constants. This theory is now valid at short times, but is more challenging to use. As an example, the diffusion-modified kinetic scheme with new connections was used to calculate kinetics of double phosphorylation and steady-state response in a phosphorylation-dephosphorylation cycle. We have reproduced the loss of bistability in the phosphorylation-dephosphorylation cycle when the enzyme reactivation time decreases, which was obtained by particle-based computer simulations. In addition to the above work on the kinetics of multisite phosphorylation, we wrote two papers with Dr. A. M. Berezhkovskii, which deal with fundamental issues in the theory of chemical reaction rates. In the first, for multidimensional diffusive dynamics, we algebraically derive 4 remarkable analytical expressions that relate the mean first passage and transition path times between two dividing surfaces with the number of unidirectional transitions per unit time (fluxes) between the two surfaces at equilibrium and the committor (the probability of reaching one surface before the other). In one dimension, such relationships can be easily derived because analytical expressions for all the above-mentioned quantities can be found. This is not possible in higher dimensions, and at first sight, the problem seems intractable. We circumvent the difficulty that the equations determining the mean first passage and transition path times cannot be solved analytically by multiplying these equations by the committor, integrating both sides and finally using the divergence theorem. A byproduct of our derivation is an analytical expression for the starting point distribution over which individual first passage and transition path times must be averaged. It turns out that this distribution is not the Boltzmann one, but it has a simple physical interpretation. Finally, in another paper in this area 5, Kramers expressions for the transition rates between two basins separated by a high barrier have been rederived in a new way using Bennett-Chandler method which focuses on the time derivative of the occupation number correlation function that describes fluctuations of the basin populations at equilibrium. This derivative is infinite at t=0 for diffusive dynamics. We show that on a slightly longer time scale, comparable to the time it takes the system to fall off the barrier, this time derivative is proportional to the spatial derivative of the committor evaluated at the barrier top. The committor or splitting probability is the probability that the system starting on the barrier ends up in one basin before the other. This probability can be found analytically. By asymptotic evaluation of the relevant integrals, we recover Kramers result without having had to rely on his formidable physical intuition.