V6: Chemical Kinetics & Transition States

V6: Chemical Kinetics & Transition States • see chapter 19 in book of K. Dill • Aim: describe kinetics of processes on energy landscapes • (e.g. chemical reactions). • temperature effect • detailed balance • mass action law • Arrhenius plots • concept of transition state/activation barrier  transition state theory • -value analysis revisited • effect of catalysts Optimization, Energy Landscapes, Protein Folding

Reaction rates are proportional to concentrations Lets consider a simple kinetic process, the interconversion between 2 states, kfand kr : forward and reverse rate coefficients. How do the amounts of A and B change with time t, given the initial amounts at time t = 0 ? The two equations are coupled. One can solve them by matrix algebra … Optimization, Energy Landscapes, Protein Folding

Excursion: coupled differential equations We can use the method of diagonalisation to solve coupled ordinary differential equations. For example, let x(t) and y(t) be differentiable functions and x' and y' their derivatives. The differential equations are relatively difficult to solve: • but u' = ku for a constant k is easy to solve. • It has the solution u = Aekx where A is a constant • Remembering this fact, we translate the ODEs into matrix form By diagonalizing the square matrix, we get www.algebra.com Optimization, Energy Landscapes, Protein Folding

Excursion: coupled differential equations By diagonalizing the square matrix, we get We then put It follows that Thus The solutions of this system are found easily: with some constants C and D. With Optimization, Energy Landscapes, Protein Folding

Reaction rates are proportional to concentrations With this technique, we could solve our system of coupled diff. equations. If kr << kf, the first equation simplies to If [A(t)] + [B(t)] = constant, then Optimization, Energy Landscapes, Protein Folding

At equilibrium, rates obey detailed balance The principle of detailed balance says that the forward and reverse rates must be identical for an elementary reaction at equilibrium: [A]eq and [B]eq : equilibrium concentrations. To see that this is a condition of equilibrium follows from inserting into resulting in Taken from Dill book Optimization, Energy Landscapes, Protein Folding

At equilibrium, rates obey detailed balance The detailed balance condition relates the rate coefficients kf and kr to the equilibrium constant K: For more complex systems, the principle of detailed balance gives more information beyond the statement of equilibrium. For a system having more than one elementary reaction, the forward and reverse rates must be equal for every elementary reaction. For this system: Let‘s consider a 3-state mechanism with kIA 0, kBI 0, kAB 0. Taken from Dill book Optimization, Energy Landscapes, Protein Folding

At equilibrium, rates obey detailed balance This results in the mechanism shown right. The only conditions for equilibrium are: Taken from Dill book These are two independent equations for 3 unknown concentrations  the system has an infinite number of solutions. In mechanism (b), all rates of the Denominator in are zero  mechanism (b) is impossible. Optimization, Energy Landscapes, Protein Folding

At equilibrium, rates obey detailed balance The principle of detailed balance says that forward and backward reactions at equilibrium cannot have different intermediate states. That is, if the forward reaction is A  I  B, the backward reaction cannot be B  A. The principle of detailed balance can be derived from microscopic statistical mechanics. Optimization, Energy Landscapes, Protein Folding

The mass action laws describe mechanisms in chemical kinetics Suppose the following reaction leading from reactants A, B, and C to product P: In general, the initial reaction rate depends on - the concentrations of the reactants - the temperature and pressure - and on the coefficients a, b, and c. Kinetic law of mass action (CM Guldberg & P Waage, 1864): „the reactants should depend on on stoichiometry in the same way that equilibrium constants do“. Although mass action is in agreement with many experiments, there are exceptions. These require a quantum mechanical description. Optimization, Energy Landscapes, Protein Folding

Reaction rates depend on temperature Consider a binary reaction in the gas phase: Suppose that By definition, the rate coefficient k2 is independent of [A] and [B]. But k2 can depend strongly on temperature. The observed dependence of the reaction rate on the temperature is much greater than one would expect from just the enhanced thermal motions of the molecules. Optimization, Energy Landscapes, Protein Folding

Arrhenius equation 1889, S. Arrhenius started from the van‘t Hoff equation for the strong dependence of the equilibrium constant K on temperature: and proposed that kfand kralso have van‘t Hoff form where Ea and E‘ahave units of energy that are chosen to fit exp. data. Eaand E‘aare called activation energies. Optimization, Energy Landscapes, Protein Folding

Activation energy diagram According to Arrhenius, it is not the average energy of the reactants that determines the reaction rates but only the high energies of the ‚activated‘ molecules. There are two plateaus, one for the reactants and one for the products. In between lies an energy maximum (also: transition state or activation barrier) which is the energy that activated molecules must have to proceed from reactants to products. Measuring kfas a function of temperature, and using eq. (1) gives Ea. Measuring the reverse rate gives E‘a. Measuring the equilibrium constant versus temperature gives h°. Taken from Dill book Optimization, Energy Landscapes, Protein Folding

Population at different temperatures From it follows The figure shows how activation is interpreted according to the Boltzmann distribution law: a small increase in temperature can lead to a relatively large increase in the population of high-energy molecules. Taken from Dill book Optimization, Energy Landscapes, Protein Folding

Arrhenius plots H2 + I2 2HI (open circles) 2HI  H2 + I2 (full circles) The figures show examples of chemical systems showing Arrhenius behavior. Taken from Dill book Integrating over temperature T gives: Diffusion of carbon in iron Optimization, Energy Landscapes, Protein Folding

Activated processes Arrhenius kinetics applies to many physical and chemical processes. When should one treat a process as activated? If a small increase in temperature gives a large increase in rate, a good first step is to try the Arrhenius model. E.g. breaking of bonds. Counter example: highly reactive radicals. These can be much faster than typical activated processes and they slow down with increasing temperature. We now describe a more microscopic approach to reaction rates, called transition state theory. Optimization, Energy Landscapes, Protein Folding

The energy landscape of a reaction An energy landscape defines how the energy of a reacting system depends on its degrees of freedom. E.g. A + BC  AB + C Each reaction trajectory would involve some excursions up the walls of the valleys. When averaged over multiple trajectories, the reaction process can be described as following the lowest energy route, along the entrance valley over the saddle point and out of the exit valley, because the Boltzmann populations are highest along that average route. Energy surface for D + H2 HD + H The transition (saddle) point is denoted by the symbol ‡. It is unstable: a ball placed on the saddle point will roll downhill along the reaction coordinate. Taken from Dill book Optimization, Energy Landscapes, Protein Folding

Calculating rate coefficients from TST Let us consider the reaction by transition state theory: Divide the reaction process into two stages: (1) the equilibrium between the reactants ant the transition state (AB)‡ with ‚equilibrium constant‘ K‡ (2) a direct step downhill from the TS to the product with rate coefficient k‡: Key assumption of TST: step (1) can be expressed as an equilibrium between the reactants A and B and the transition state (AB)‡ , with even though (AB)‡ is not a true equilibrium state. Optimization, Energy Landscapes, Protein Folding

Calculating rate coefficients from TST The overall rate is expressed as the number of molecules in the TS, [(AB)‡], multiplied by the rate coefficient k‡ for the second product-forming step Because the quantitiy K‡ is regarded as an equilibrium constant, it can be expressed in terms of the molar partition functions: where D‡ is the dissociation energy of the TS minus the dissociation energy of the reactants. q(AB)‡ is the partition function of the transition state. Optimization, Energy Landscapes, Protein Folding

The transition state (left) Contour plot of a reaction pathway (- - -) on an energy landscape for the reaction A + BC  AB + C. The broken line shows the lowest-energy path between reactants and products (right) The transition state is an unstable point along the reaction pathway (indicated by the arrow) and a stable point in all other directions that are normal to the reaction coordinate. Taken from Dill book Optimization, Energy Landscapes, Protein Folding

Calculating rate coefficients from TST Provided that the unstable degrees of freedom are independent of the stable degrees of freedom, q(AB)‡ is factorable into two components represents the partition function for all of the ordinary thermodynamic degrees of freedom of the TS structure, and q represents the one nonequilibrium vibrational degree of freedom of the bond along the reaction coordinate. The partition function of a vibration is for a weak vibration with low frequency . Once the system has reached the transition state, it is assumed to proceed as quickly to the product state as the system permits, namely at the frequency of the reaction coordinate vibration, Optimization, Energy Landscapes, Protein Folding

deviations from TST A factor , the transmission coefficient, is often introduced (k‡ =  ) to account for observed deviations from the simple rate theory. In condensed-phase medium, or in complex systems,  < 1. This gives the relation between the rate coefficient k2and the partition functions: Optimization, Energy Landscapes, Protein Folding

relation between  value analysis and TST In V5, we characterized the effect of a protein mutant by its -value G0 reflects whether the mutant stabilizes the folded state F over the unfolded state U stronger or weaker than wild-type protein. According to TST, both wild-type and mutant folding proceed via transition states with activation free energies G‡wtand G‡mut. A -value of 1 means that G0 = G‡ for this mutant  the mutant has the same effect on the TS structure as on the folded state  this part of the TS structure is folded as in the folded state F. Optimization, Energy Landscapes, Protein Folding

Catalysts speed up chemical reactions Catalysts affect the rates of chemical reactions; e.g. enzymes accelerate biochemical reactions. Enzymes can achieve remarkable accelerations, e.g. by a factor of 2 x 1023 for orotine 5‘-phosphate decarboxylase. Linus Pauling proposed in 1946 that catalysts work by stabilizing the transition state. kc ABC Linus Pauling 1935 Free energy barrier G‡ is reduced by a catalyst C. Taken from Dill book Optimization, Energy Landscapes, Protein Folding

Catalysts speed up chemical reactions From transition theory we obtain for the catalyzed reaction rate kc (normalized to the uncatalyzed reaction rate k0) This ration represents the ‚binding constant‘ of the catalyst to the transition state  the rate enhancement by the catalyst is proportional to the binding affinity of the catalyst for the transition state. This has two important implications: (1) to accelerate a reaction, Pauling‘s principle says to design a catalyst that binds tightly to the transition state (and not the reactants or product, e.g.). (2) a catalyst that reduces the transition state free energy for the forward reaction is also a catalyst for the backward reaction. Optimization, Energy Landscapes, Protein Folding

Speeding up reactions by intramolecular localization or solvent preorganization Reactants polarize, so water reorganizes. Two neutral reactants become charged in the transition state. Creating this charge separation costs free energy because it orients the solvent dipoles. Taken from Dill book Optimization, Energy Landscapes, Protein Folding

Speeding up reactions by intramolecular localization or solvent preorganization Enzymes can reduce the activation barrier by having a site with pre-organized dipoles. Taken from Dill book Optimization, Energy Landscapes, Protein Folding

Funnel landscape describe diffusion and polymer folding All the processes described sofar involve well-defined reactants and products, and a well-defined reaction coordinate. But diffusional processes and polymer conformational changes often cannot be described in this way. The starting point of protein folding is not a single point on an energy landscape but a broad distribution. A bumpy energy landscape, such as occurs in diffusion processes, polymer conformational changes, and biomolecule folding. A single minimum in the center may represent ‚product‘, but there can be many different ‚reactants‘, such as the many open configurations of a denatured protein. http://www.dillgroup.ucsf.edu/ Optimization, Energy Landscapes, Protein Folding

Summary Chemical reactions and diffusion processes usually speed up with temperature. This can be explained in terms of a transition state or activation barrier and an equilibrium between reactants and a transient, unstable transition state. For chemical reactions, the transition state involves an unstable weak vibration along the reaction coordinate, and an equilibrium between all other degrees of freedom. Catalysts act by binding to the transition state structure. They can speed up reactions by forcing the reactants into transition-state-like configurations. Optimization, Energy Landscapes, Protein Folding

V6: Chemical Kinetics & Transition States