Riesgo de mercado

Consider a physical system with two well defined states, A and B, and some path- way enabling a transition between them, for example, a standard, reversible chemical reaction. It is often the case that the full mechanism by which the reaction occurs is highly complex, yet the information required from a study of the reaction doesn’t require knowledge of the mechanism at all. For example, for a given amount of re- actant one may simply wish to determine the amount of product that will be formed given certain environmental conditions. In this situation we may ignore the specific atomistic processes involved over the course of the reaction and simply use a kinetic model, which uses the statistics of equilibrated systems to determine the end result of a given reversible reaction. All that is required is prior knowledge of the reaction rates, which can be measured via experimentation.

4.2.1 A Chemical Example - Table Salt

As a practical example, take the traditional example given in basic chemistry lessons for generating table salt,

NaOH + HCl −−*)−− NaCl + H2O. (4.1)

Although the mechanism for the reaction is known, for the sake of example let us assume for now that it is not. Regardless, we are simply be interested in how much product we will obtain given an initial amount of reactant together with some set of environmental conditions. For a given reaction that has reached equilibrium, we can define the presence of reactant and the presence of product as separate, discrete kinetic states. From Equation4.1, we will define state A as NaOH+HCl and state B as

Chapter 4. Kinetic Scheme 81

NaCl+H₂O. If we assume stoichiometry in the reaction, the equilibrium concentrations of both states can then be obtained using the principle of detailed balance as follows, cARAB = cBRBA, (4.2)

where cA and cB are the equilibrium concentrations of states A and B respectively,

and due to our (assumed) lack of knowledge on the specific molecular mechanisms occurring during the reaction, we are required to include reaction probabilities in the form of statistical averages, the equilibrium reaction rates RAB and RBA, which are

approximately constant at equilibrium. In actuality, the rates are a function of the specific molecular details and are therefore typically related to an energy barrier be- tween states. Environmental conditions such as temperature, pressure, and presence of catalysts, as well as reactant concentrations and other factors all contribute to the rates by affecting the ability of the molecular ensembles to acquire enough energy to proceed in the reaction. But in principle for our study of interest, the specific molecu- lar details can be bypassed in favour of the reaction rates, which contain the net result of all of the higher resolution detail and can be measured directly through relatively simple experimentation.

Once the rates are determined, we can solve Equation 4.2 to determine the equilib- rium concentrations of both reactant and product. We can assume that the total concentration of initial mixture is known, such that,

cA+ cB = cT, (4.3)

where cT is the total initial concentration of both product and reactant. Substitution

into Equation 4.2gives,

cA= RBA RAB+ RBA cT, cB = RAB RAB+ RBA cT. (4.4) 4.2.2 Kinetic Networks

Kinetic theory can be generalised to any number of discrete states forming a connected network. At equilibrium, for any two directly connected states within the network, indexed by i and j, detailed balance requires that,

Chapter 4. Kinetic Scheme 82

Figure 4.1: A visualisation of a generic kinetic network, where the occupancies of each state are visualised by their areas. The detailed balance condition is applicable to every pair of connected states in this network. Although each transition is reversible, we can see a clear inhomogeneity in the occupancies, implying that the transition

rates towards state 2 are large, and those towards state 4 are small.

Such a network is illustrated in Figure4.1.

The reaction rates within these networks can often be individually determined via ex- perimentation. The complete set of rates means that the network corresponds to a fully parametrised Markov chain of events [99] for which we can determine the equi- librium distribution of concentration around the whole network using methods similar to Equation 4.4.

The idea of detailed balance can also be applied to the kinetics of single molecules. By dividing throughout by the total concentration, cT, we can rewrite Equation 4.5in

terms of occupation probabilities,

PiRij = PjRji, (4.6)

where Pi is the probability of being in state i at any given time once equilibrium is

reached. For our example network, this represents having only a single state occupied within the system at a time. We can see how this kind of model can be useful by looking at studies of molecular motors.

Chapter 4. Kinetic Scheme 83

4.2.3 A Biological Example - Molecular Motors

A set of molecules present within biological organisms responsible for generating force are the molecular motors. Molecular motors feel the effect of thermal fluctuations as all proteins do, and as a result dynamically explore conformational space about a well defined equilibrium configuration. The generation of force is not due to these fluctuations however, but due to internally metabolised ATP providing enough energy to change the equilibrium state itself. These large transitions between equilibrium structures enable the motor to do very specific work against any attached cargo and it is this cycle that generates force and the useful work done by the molecule. Both of these equilibrium structures have been solved to atomic resolution for a variety of motors but what is still largely unknown is exactly how the transitions proceed at the atomic level.

With incomplete structural detail regarding the transition process, a kinetic model is suitable for studying a molecular motor, and indeed, a purely kinetic model of the motor cytoplasmic dynein has been designed by Zhao et al. [100]. Cytoplasmic dynein is a highly complex dimer known to be responsible for intra-cellular transport of vital cargoes via a bipedal walking mechanism, with each step driven by an ATP hydrolysis fuelled ‘power-stroke’. The ‘pre’ and ‘post’ power-stroke equilibrium structures were modelled by Zhao as discrete kinetic states. With very little ab initio knowledge of the underlying transition mechanism from pre-powerstroke to post-powerstroke or vice versa, their model is able to calculate a probability distribution of the step size (including direction) of the motor, which indicated a bias towards forward motion with the number of ‘hand over hand’ steps1 being ∼ 4 times less prevalent than so-called ‘inchworm’ steps2. The authors make a number of assumptions regarding the rates, yet we can still see how powerful a kinetic model is when parametrised only by a small number of transition rates. Each rate can be easily modified within the model to see the global effect of altering the rate at which processes occur, such as ATP hydrolysis (by varying the ATP concentration), or diffusion (by varying the background viscosity). We will perform our own study of cytoplasmic dynein in more detail in Chapter5. Following our discussion and development of the dynamical FFEA model in Chapter3, one might consider whether the thermal fluctuations may have a significant effect on these reaction rates. In cytoplasmic dynein, for example, the structural deformations at equilibrium are relatively large and could therefore undergo associated energy changes that are potentially comparable to that released via ATP hydrolysis. If this is the case,

1

Steps in which the trailing monomer moves past the leading monomer, or vice versa

2

Steps in which the trailing monomer merely catches up to the leading monomer, or vice versa, without overtaking.

Chapter 4. Kinetic Scheme 84

then the effect of these fluctuations on the kinetic transition rates of single molecules should be considered. The remainder of this chapter concerns the development of a coupled dynamic-kinetic model and its implementation within the FFEA framework.

4.2.4 A Note on Nomenclature

We define the ‘kinetics’ we have introduced in this chapter as the modelling of occupa- tion probabilities of discrete states defined within a system, and the rates of transition between them. We explicitly distinguish this type of modelling from the inclusion of ‘dynamics’, by which we mean that the entire trajectory of the system is calculated from underlying, fundamental laws of motion. These two models are mutually exclu- sive when simulating a specific process, in the sense that if one has enough knowledge about the underlying physical transition process to determine appropriate equations of motion, then the dynamical model can replace the kinetic model. What we can do is use these models in conjunction with one another to retain the advantages of a low spatial resolution dynamical simulation (FFEA, for example) whilst applying kinetic theory to model discrete, atomistic events in a statistical manner, without the need for the inclusion of high resolution detail.