1.2. Chemical kinetics in time-resolved crystallography
Chemical kinetics (Steinfeld et al.
) describes a reaction in terms of schemes similar to the ones shown in Fig. 1, where a mechanism is depicted that employs only first-order reactions. Each step is characterized by its respective rate coefficient. Rate coefficients cannot be measured directly. Instead, macroscopic rates (or their negative inverse, the relaxation times) are directly observable in kinetic measurements. They are functions of all rate coefficients (Matsen & Franklin, 1950
; Fleck, 1971
). It is important to make a clear-cut distinction between experimentally observable (macroscopic) rates and the hidden (microscopic) rate coefficients of the mechanism (Rajcu et al.
). It is the purpose and the goal of any kinetic experiment to ultimately determine the underlying kinetic mechanism with all of its rate coefficients. Almost any kinetic model is mathematically underdetermined if relaxation times from a time series at only one temperature are available, because a large number of rate coefficients must be determined from a smaller number of measured relaxation times. Since a time-resolved crystallographic experiment is a true kinetic experiment, it depends on the kinetic mechanism in fundamentally the same fashion.
Figure 1 Kinetic mechanisms. General: general mechanism with three intermediates I
2 and I
3, plus the reference (dark) state. The reaction is started by a laser pulse. DE: dead-end candidate, SP: semi-parallel (more ...)
As an example we simulated relaxation times similar to those of a recent time-resolved crystallographic experiment (Schmidt et al.
). Three relaxation times are observed. The number of relaxation times is equal to (or smaller than) the number of intermediates. A general chemical kinetic mechanism of a cyclic reaction with three intermediates plus the final (dark) state is shown in Fig. 1. The general mechanism contains 12 rate coefficients (that from I
directly to D
is not shown). It is immediately clear that the three relaxation times are not sufficient to uniquely determine all rate coefficients of the general mechanism. Therefore, two likely candidate mechanisms, each involving four rate coefficients, were picked in a rather subjective way (DE: dead-end candidate, SP: semi-parallel candidate). Fig. 2(a
) shows that the rate coefficients for the two likely mechanisms can be such that the time-dependent concentrations of the intermediates match (almost) exactly at a certain temperature, here 300 K. At this temperature these two mechanisms are degenerate, because they give the same relaxation times. Now assume that in the DE mechanism intermediate I
is very stable. That is, the barrier of activation to revert to intermediate I
is quite high. Then, the rate coefficient k
becomes particularly small at lower temperatures. On the other hand, in the semi-parallel mechanism SP, intermediate 3 may branch directly to the final (dark) state, crossing a smaller energy barrier. If the temperature is lowered to, say, 273 K, these two mechanisms become separable (Fig. 2b
), although they were indistinguishable at 300 K. Consequently, the temperature adds observables that can be used to determine the unknowns (the rate coefficients) of the mechanisms.
Figure 2 Concentration profiles and approximate relaxation times for the three intermediates in the semi-parallel and dead-end mechanisms. (a) 300 K, semi-parallel mechanism: intermediate I
1 (dashed line), intermediate I
The rate coefficients at two temperatures are related. In the simplest case this relationship is given by the Arrhenius equation (see, for example, Cornish-Bowden, 1999
), where k
is the Boltzmann constant and T
is the temperature,
For each rate coefficient k
we would need to determine three parameters: the enthalpy (ΔH
) and entropy (ΔS
) differences to the transition state, as well as a pre-factor A
). In the simplest case, A
) is proportional to the temperature (see also Cornish-Bowden, 1999
). Hence, the fit would include a linear term, a constant and an exponential term. If we were able to determine experimentally only the relaxation times, we would need 12 different temperatures to account for the 36 free parameters in the general mechanism of this example. If we were only interested in the ΔG
values in equation (1)
, we would need eight temperatures (24 free parameters). The critical question is whether measurement over a limited physiological temperature range would allow us to determine all the unknowns from measured relaxation times.
However, we are in fact considerably better off, because we can exploit the so-called absolute scale present in crystallography. Measured structure factor amplitudes can always be placed on an absolute scale by scaling them to the calculated structure factor amplitudes from a precise known structural model. Electron density, when on an absolute scale, directly relates to occupancy; that is, to fractional concentration. For example, if a side chain has alternate conformations, the electron density is directly related to the occupancy of that conformation. The same is true for difference electron densities on the absolute scale. In contrast, in optical absorption spectroscopy, for example, absorption usually does not directly relate to concentration as there is a linear factor, the absorption coefficient, which is unknown a priori for various intermediates. In crystallography this linear factor is simply absent. Consider an experiment in which a CO molecule is photo-dissociated from the iron of a heme protein. By integrating the negative difference electron density at the CO binding site, seven electrons are obtained. Since CO has 14 electrons, 50% of the CO has been photo-dissociated. If the concentration of protein is 50 mM in the crystal, the remaining bound CO concentration is 25 mM (50% of the 50 mM). Note that this calculation is only valid on the absolute scale.
Now, we need to connect the structural result, the electron density, to the kinetics. Equation (2)
is the result of integrating the coupled differential equations that describe the mechanism involving three intermediates assuming exponential kinetics. Equation (2)
constitutes the mathematical base of simple chemical kinetics described in textbooks such as Steinfeld et al.
). The time-dependent concentrations [I
] can be calculated from the rate coefficients, because both the relaxation times τk
, which are the eigenvalues of the so-called coefficient matrix, and the pre-factors [P
], which are the elements of the eigenvectors of the coefficient matrix, are functions of the rate coefficients; the [P
] contain further specific initial conditions, for example [I
] = 1.0, [I
] = [I
] = 0 at t
Now assume that we are able to observe the [P
] directly. That would give us, in the best scenario, another nine independent observables to determine the underlying rate coefficients. However, some of the [P
] could be very small, close to zero, because each intermediate might have only one transient. In our example in Fig. 2, [P
] and [P
] are in fact zero (and some others are also negligible in magnitude and not independent). This reduces the number of useful [P
]. In any case, we would obtain at least three additional observables if the [P
] could be observed. And indeed, the [P
] can be observed directly in crystallography. How can that be done? This approach has been the foundation of the ‘posterior analysis’ developed by Schmidt et al.
) for post-SVD analysis of time-resolved data. Details are given also by Schmidt et al.
) and Schmidt (2008
). In short, once the preliminary structures of the intermediates are determined, we can calculate time-independent difference electron densities for each intermediate by subtracting the structure factors of the known initial (reference) D
state from those of the intermediates. We assume a chemical kinetic mechanism with initial values for the rate constants and generate with these calculated time-dependent difference electron densities
that can be compared directly with the observed difference densities
. In a large fitting routine the rate coefficients and the initial condition, namely the concentration of activated molecules at the beginning of the reaction, are refined by minimizing globally the difference between the observed and calculated difference maps. Both the amplitudes [P
] and the measured relaxation times τj
are used as observables this way to refine the numerical values of the rate coefficients k
. The fact that concentration is directly related to electron density enhances greatly the ability to determine a comprehensive chemical kinetic mechanism from time-resolved crystallographic data.
With the relaxation times τk and the amplitudes [P
jk] we gain at least six observables per temperature to determine the 36 free parameters (plus one initial condition, the extent of reaction initiation) in the general mechanism. That would reduce the number of required temperatures to only six or seven. It is extremely exciting to see that we actually have a realistic chance to determine a comprehensive general mechanism with measurements at a relatively small number of temperatures. However, the collection of an extensive spatially and temporally complete time series of Laue data is a tedious time-consuming process let alone the collection of time series at multiple temperatures. It is a major goal of this paper to show that this can now be achieved.