Diffusion limited molecular collisions

For a soluble enzyme that is not part of a multi-enzyme complex, the fastest rate of enzyme-inhibitor association is determined by the rate of molecular collisions between the two binding partners (i.e., the enzyme and the inhibitor) in solution. The rate of molecular collisions is in turn controlled by the rate of diffusion. The diffusion-limited rate of molecular collisions is dependent on the radii of the two binding molecules and the solution temperature and viscosity (Fersht, 1999) ... 

From the above outline, the mass-transport problem is seen to consist of coupled boundary value problems (in gas and aqueous phase) with an interfacial boundary condition. Cloud droplets are sufficiently sparse (typical separation is of order 100 drop radii) that drops may be treated as independent. For cloud droplets (diameter 5 ym to 40 pm) both gas- and aqueous-phase mass-transport are dominated by molecular diffusion. The flux across the interface is given by the molecular collision rate times an accommodation coefficient (a 1) that represents the fraction of collisions leading to transfer of material across the interface. Magnitudes of mass-accommodation coefficients are not well known generally and this holds especially in the case of solute gases upon aqueous solutions. For this reason a is treated as an adjustable parameter, and we examine the values of a for which interfacial mass-transport limitation is significant. Values of a in the range 10 6 to 1 have been assumed in recent studies (e.g.,... 

When mass transfer to the catalyst surface is fast compared to chemical reaction, the reaction rate may still be limited by the rate of diffusion in the catalyst pores. Bulk diffusion (i.e. the same process as for mass transfer to the external surface) occurs when the mean free path between molecular collisions is small compared to the pore radius. Knudsen diffusion occurs when the mean free path is large... 

Three thousand separate trajectories were simulated for each receptor density and at each truncation height. In contrast to molecular dynamics, in BDS the solvent is treated as a viscous continuum and, hence, time steps on the order of 0.1 ps are appropriate. Diffusion-limited binding or instantaneous reaction upon ligand-receptor collision was assumed. Computationally, this condition was considered met when the center of the ligand particle and the center of the receptor were less than 20 A apart. Trajectories were truncated when the ligand s height above the membrane surpassed Q, the truncation distance. Sample trajectories of both a successful capture and an escape are shown in Figs. 16a and 16b. 

We might ask how fast can a reaction possibly go In other words, if every molecular collision resulted in a reaction, how fast would that be This is known as the diffusion limit, and it provides a useful number with which to compare what we see in practice. 

The rates of diffusion-limited termolecular elementary processes in liquid phases are proportional to the number of encounter pairs (pairs of molecules in the midst of an encounter) and also to the number of third molecules present to diffuse into the same cage as the encounter pair. Therefore, diffusion-limited termolecular elementary processes are third order, just as in the gas phase. Activation-limited termolecular elementary reactions in liquids are also third order if the fraetion of collisions that lead to reaction is independent of the concentration. We assume further that unimolecular elementary processes in liquids exhibit first-order kinetics, as in the gas phase. We can now summarize the facts for elementary processes in both liquids and gases The molecularity of a substance in an elementary process is equal to its order, and the overall order is equal to the sum of the orders of the individual substances. 

Straub J E and Berne B J 1986 Energy diffusion in many dimensional Markovian systems the consequences of the competition between inter- and intra-molecular vibrational energy transfer J. Chem. Phys. 85 2999 Straub J E, Borkovec M and Berne B J 1987 Numerical simulation of rate constants for a two degree of freedom system in the weak collision limit J. Chem. Phys. 86 4296... 

Although long-time Debye relaxation proceeds exponentially, short-time deviations are detectable which represent inertial effects (free rotation between collisions) as well as interparticle interaction during collisions. In Debye s limit the spectra have already collapsed and their Lorentzian centre has a width proportional to the rotational diffusion coefficient. In fact this result is model-independent. Only shape analysis of the far wings can discriminate between different models of molecular reorientation and explain the high-frequency pecularities of IR and FIR spectra (like Poley absorption). In the conclusion of Chapter 2 we attract the readers attention to the solution of the inverse problem which is the extraction of the angular momentum correlation function from optical spectra of liquids. 

The orientation of linear rotators in space is defined by a single vector directed along a molecular axis. The orientation of this vector and the angular momentum may be specified within the limits set by the uncertainty relation. In a rarefied gas angular momentum is well conserved at least during the free path. In a dense liquid it is a molecule s orientation that is kept fixed to a first approximation. Since collisions in dense gas and liquid change the direction and rate of rotation too often, the rotation turns into a process of small random walks of the molecular axis. Consequently, reorientation of molecules in a liquid may be considered as diffusion of the symmetry axis in angular space, as was first done by Debye [1],... 

Another way of evaluating enzymatic activity is by comparing k2 values. This first-order rate constant reflects the capacity of the enzyme-substrate complex ES to form the product P. Confusingly, k2 is also known as the catalytic constant and is sometimes written as kcal. It is in fact the equivalent of the enzyme s TOF, since it defines the number of catalytic cycles the enzyme can undergo in one time unit. The k2 (or kcat) value is obtained from the initial reaction rate, and thus pertains to the rate at high substrate concentrations. Some enzymes are so fast and so selective that their k2/Km ratio approaches molecular diffusion rates (108—109 m s-1). This means that every substrate/enzyme collision is fruitful, and the reaction rate is limited only by how fast the substrate molecules diffuse to the enzyme. Such enzymes are called kinetically perfect enzymes [26],... 

University, Krak6w [i]. He described Brownian molecular motion independently from Einstein considering the collisions explicitly between a particle and the surrounding solvent molecules [ii], worked on colloids [iv-v], and obtained an expression for the rate with which two particles diffuse together (-> Smoluchowski equation (b)) [iii-v]. He also derived an equation for the limiting velocity of electroosmotic flow through a capillary (-> Smoluchowski equation (a)). 

The general principle of BD is based on Brownian motion, which is the random movement of solute molecules in dilute solution that result from repeated collisions of the solute with solvent molecules. In BD, solute molecules diffuse under the influence of systematic intermolecular and intramolecular forces, which are subject to frictional damping by the solvent, and the stochastic effects of the solvent, which is modeled as a continuum. The BD technique allows the generation of trajectories on much longer temporal and spatial scales than is feasible with molecular dynamics simulations, which are currently limited to a time of about 10 ns for medium-sized proteins. 

For noninteracting particles D b is + D, but as the particles approach each other, the relative diffusion coefficient becomes dependent on their spatial separation. In liquids for large particles this arises from hydrodynamic interactions ( bow waves ), while in the gas phase the particles screen each other from the bath collisions. For small particles the viscoelastic projjerties of the fluid will become important near contact. The solution of Eq. (2.23) applies only for sufficiently large friction where the relative motion on all length scales is diffusive. In the other limit of very low friction, the general result obtained from molecular theory is of the form... 

Vibrational relaxation and excitation and usually the rate-limiting processes for molecular reaction in the gas phase, and their importance has led to many theoretical approaches. The use of an FPE such as described in Section II leads to a diffusion model in energy space, and only applies if the collision kernel P(E, E ) of the master equation is strongly peaked about the initial energy E. This is the weak collision limit in which the energy transfer is small, or comparable to kT. Other approaches, such as the model of Bhatnager, Gross and Krook propose that impulsive collisions randomize... 

In the first approximation the collision phenomena are described in terms of hard sphere molecular diameters, which are independent of temperature. Actually, the diameters decrease with higher temperature, approaching individual limits [1]. Let us consider a single molecular entity with the mass m and the diameter afmj, which diffuses through a gas consisting mainly of more abundant dissimilar molecules of the mass m2, the diameter dm.2 and the concentration no. If the collision diameter is mi.2 = (dm, 1 + <7m.2)/2, the tracer molecule must collide each second with the host molecules contained in a volume of about nm 2um. Because the host molecules also move, the mean relative speed u 1,2 is... 

SECTION 10.8 It follows from Idnetic-molecular theory that the rate at which a gas undergoes effusion (escapes through a tiny hole) is inversely proportional to the square root of its molar mass (Graham law). The diffusion of one gas through the space occupied by a second gas is another phenomenon related to the speeds at which molecules move. Because molecules undergo frequent collisions with one another, the mean fiee path—the mean distance traveled between collisions- -is short. Collisions between molecules limit the rate at which a gas molecule can diffuse. 

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