Ligand diffusion in globins: Simulations versus Experiment

Summary

Computer simulations in molecular biophysics describe in atomic detail structure, dynamics, and function of biological macromolecules. To assess the quality of these models and to pick up new mechanisms, comparisons with experimental measurements are made. Most comparisons examine thermodynamic and average structural properties. Here we discuss studies of dynamics and fluctuations in a protein. The diffusion of a small ligand between internal cavities in myoglobin, and its escape to solvent are considered. Qualitative and semi-quantitative agreements between experiment and simulation are obtained for the identities of the cavities that physically trap the ligand and for the connections between them. However, experimental and computational “doors” are at significant variance. Simulations suggest multiple gates while kinetic experiments point to one dominant exit.

Introduction

After the determination of the globin structures by X-ray crystallography[1, 2] it was quickly established that thermal fluctuations of the protein matrix must assist the penetration of the small ligand in and out of the protein. Visual inspection of the X-ray structure suggests no obvious way for a ligand to enter or escape. Indeed a pioneering computational work by Case and Karplus [3] examined two routes between the binding site and the solvent just below the E helix (figure 1) exiting through the so-called histidine gate. This study illustrates significant energy barriers for the “door” and a mechanism in which side chain fluctuations are coupled to ligand escape and to partial protein fluidity [4]. It is amusing that this particular path, after significant debate and suggestions for alternatives in the literature [5, 6], recently received a strong endorsement from experimental studies [7, 8]. A continuous flow of theoretical and experimental studies of this system followed these original investigations. This paper provides an overview on the field with a focus on recent computational investigations. Other influential papers from the past to which frequent references are made are (i) the identification of the Xenon cavities in the protein structure and the use of a small computational probe to study them [5], and (ii) the first systematic search of diffusion pathways of a diatomic ligand in a thermally fluctuating protein matrix [6].

Figure 1.

Myoglobin and the Xenon cavities (detected by free volume calculations in red). The heme is in blue and the red spot attached to it from above is the distal pocket. Above the distal pocket one finds the Xe4 cavity and below the heme is the Xe1 binding site. Xe2 and Xe3 are to the left. The E helix is drawn in the back from the lower left to the upper right of the protein, touching in the figure the cavities Xe2-Xe4. The figure was prepared by the program zmoil http://clsb.ices.utexas.edu/prebuilt/.

Ligand diffusion into and from buried active sites is of wide interest. The Locally Enhanced Sampling (LES) method, for searching diffusion pathways, was introduced for myoglobin [6]. However, it was used to investigate other proteins, for example, the diffusion of oxygen and hydrogen in hydrogenase [9], of oxygen in dioxygenase [10], and of nitric oxide in nitrile hydratase [11]. Interestingly, more accurate variant of LES is available [12, 13], but so far it was not used widely. Other examples in which Molecular Dynamics provided insight to ligand permeation problems are of flavoenzyme [14], and catalase [15].

In addition to a deeper understanding of molecular biophysics events that are relevant to molecular function, thermally assisted diffusion is an ideal system for a straightforward study by Molecular Dynamics (MD). The mantra for accurate application of the MD approach is (i) force field and (ii) sampling. Both items seem in place for the task at hand. The forces involved are physical, no formation or breaking of chemical bonds are present. This observation suggests that traditional and readily available mechanical force fields are a reasonable match.

The sampling requirement is somewhat more complex. The time scale of the events of interest (migration of the ligand to alternative cavities in the protein and to the solvent) is pretty long compared to the basic time step of MD (femtoseconds). Ligand hops between cavities and escapes at time scales measured in hundreds of nanoseconds [7, 8, 16]. This makes it necessary to execute hundreds of millions of time steps to observe (frequently) the events of interest. Moreover, to obtain quantitative description of ligand migration, multiple trajectories are required. The diffusion is a non-equilibrium process (once the ligand leaves the protein matrix it is unlikely to return on the simulation time scale) and an average over initial conditions must be performed to obtain quantitative results. The requirement for multiple trajectories and ensemble average makes the calculation more expensive by orders of magnitude compared to a single trajectory.

This computational task was considered formidable until recently. Past studies were therefore primarily qualitative. Even recently a few approximate approaches to the problem were explored. For example, estimates of free energy landscapes using metadynamics or particle insertion methods. Nevertheless, and as argued below, the qualitative computational picture of ligand diffusion in myoglobin did not deviate much from the earlier studies. Arguably, the major remaining qualitative puzzle of the kinetics of ligand diffusion in myoglobin is the identification of the escape pathway from the protein. At present there is a sharp disagreement between simulations and experiments (but reasonable agreement between different simulations). The source of the significant qualitative difference is not clear.

Of course considerably more work remains to be done to make the agreement between experiment and theory quantitative. Advances in software and hardware [17–19] made it possible to run a significant number of straightforward MD trajectories for substantial lengths of time. These trajectories observed migration events at room temperature that with sufficient statistics can be compared quantitatively to experiment. Hence, the results of ligand diffusion in myoglobin provide a strict test of the way we investigate kinetic mechanisms by computer simulations and are useful probes of anisotropic dynamic fluctuations in well-defined three-dimensional structures of proteins[20, 21].

Experimental analysis of ligand diffusion

Historically, the dominant experiment for studying ligand diffusion in myoglobin was of geminate recombination kinetics. The iron-ligand is broken by light or by other means and the recombination time of the dissociated ligand to the iron is followed by spectroscopy of the heme [22]. The diffusion of the ligand away from the iron and in the protein matrix causes delay in re-binding, allowing for interesting kinetic analysis. The data are interpreted in terms of a competition between (i) reformation of ligand – iron bond, (ii) diffusion between internal protein cavities (the so called DP – the distal pocket, and the Xe1-Xe4 cavities [5], see also figure 1), and (iii) escape from the protein matrix to the solvent. It is important to separate these steps both in application and analyses of theory and in experiment to make the comparison meaningful.

Other types of experiments are available that probe the ligand motions directly and help establish a more complete picture of the diffusion (e.g. time resolved crystallography [23–25] and time resolved vibrational spectroscopy [26, 27]). Significant credit to the broader and renewed interest in myoglobin kinetics is due to the spectacular snapshots in time of ligand migration between internal cavities such as the DP, Xe4 and Xe1 [5, 23–25, 28–32]. Nevertheless, studies of kinetics remain a crucial component. This is due to the relative ease of experimental kinetics, and the ability to probe ligand escape to the solvent (the time scale in which the reaction becomes concentration dependent). After all, kinetics and thermodynamics are the prime determinants of biological function. Kinetics was measured for a wide range of myoglobin mutants, and for different solution conditions such as (Xe) pressure.

We can divide the type of computations probing ligand diffusion into two main categories: (i) Approximate calculations of diffusion rate and free energy landscape, and (ii) Straightforward Molecular Dynamics trajectories. Below we discuss both of them.

Approximate calculations of diffusion rate

The approximate procedures aim at better sampling of the free energy landscape for ligand diffusion. This is achieved by making plausible assumptions on the system dynamics, assumptions that greatly simplify the computational cost of computing the free energy landscape. Such approaches are important even now in qualitative understanding of the dynamics and interpreting complex motions in a simpler form. We therefore discuss some of these ideas below. One approach of the so-called meta-dynamics [33] or local elevation [34] was used to investigate carbon monoxide diffusion in myoglobin [35, 36]. The overall diffusion patterns between myoglobin cavities were similar to other studies. Free energies of wells and of barriers between the minima were estimated. The calculated barriers vary between zero and 14.5kT in one of the studies [36], and are larger than expected from experiment. The time scale of escape to the solvent measured experimentally is about 100 nanoseconds [7, 8] while the above barrier is likely to yield longer time scales. Taking the free diffusion time scale to be τ₀ ≈ 2 ps, the activated time, τ_a, according to the above estimated barrier height is τ_a = τ₀ · exp(B/kT)≈ 1μs. Interestingly, a second meta-dynamics study [37] finds significantly lower barriers suggesting that convergence of quantitative free energy map for diffusion was not reached. For example, the barrier between the Xe4 and the Xe1 cavities in the last study, which are on opposite sides of the heme, is surprisingly small. It is estimated as 2.17 kT, corresponding to tens of picoseconds migration time at room temperature. We note that both simulations used the AMBER force field [38] so the variations are likely to be caused by different sampling. A third and recent simulation of the free energy landscape of CO diffusion in myoglobin [39] was conducted with the TAMD (Temperature Accelerated Molecular Dynamics) and the single sweep methods [40]. This time the CHARMM force field was used, and the free energy values were close to the study of Nishihara, et al. [36].

Two issues make the quantitative application of meta-dynamics difficult. The first is that the diffusion is a non-stationary process. A number of studies were conducted on relaxation processes of the protein molecule that happen simultaneously and may induce significant non-equilibrium aspects to the diffusion [21, 41–43]. As long as the ligand is trapped in the protein matrix it may be considered in local equilibrium. However, an escape to the solvent interrupts this equilibrium process. Ligands that exit to the aqueous solution may not return to the protein on the simulation time scale. Given that the computed barrier heights for ligand migration (internally and to the solvent) are not so different, e.g. 13.7 versus 14.5 kT [36], the equilibrium assumption may require further investigations. In contrast to meta-dynamics the single sweep method [39] computes rigorously the equilibrium free energy surface.

Furthermore, in meta-dynamics and the single sweep method it is necessary to decide on slow coordinates that require a “speed-up” or flattening of the energy surface. In all of the above studies the coordinates were the Cartesian position of the CO molecule. This choice is intuitive; however, it may be insufficient. Coupling between the ligand motions and activated transitions of side chains was proposed as the exit gate at the very beginning of the field [3, 44]. So far this coupling has not been demonstrated to be wrong. A slow variable based only on the ligand coordinate may be an oversimplification. Finally, the escape pathway from the Xe3 cavity proposed in reference [36] is in disagreement with experiment that strongly suggests an escape pathway from the distal pocket (DP) [7, 16]. The escape pathway(s) is at variance with experiment in essentially all simulations that searched for it, and is not specific to the meta-dynamics calculations.

Another intriguing computational probe of ligand diffusion is the treatment of the ligand as a perturbation. Plausible pathways are then determined by the static structure and protein fluctuations computed without the ligand. The free energy landscape is estimated by one-step insertion of the ligand to pre-existing tunnels within the protein matrix [45–47]. The calculations are very efficient and make it possible to probe different globins and their alternate internal diffusion network. The underlying assumption is that the small (but hard) ligand does not significantly push protein atoms while moving around.

Straightforward Molecular Dynamics

Given that different approximate theories (and even the same theory) provide quantitatively different results it is necessary to re-evaluate the basic model (i.e. the force field) as well as the approximations that are part of the above theories. From this perspective it is useful to perform the most straightforward calculation possible in which approximations are eliminated allowing for better evaluation of agreement (or disagreement) between simulations and experiments.

For diatomic molecules such as oxygen or carbon monoxide molecules the prime interaction with other protein atoms (or solvent molecules) is of excluded volume. This interaction is typically modeled as a Lennard Jones potential. Small dipole (or more significant) quadrupole moments are used less frequently, even though well-tested parameterization is available [48, 49]. The overall diffusion pattern of the diatomic ligand in the protein matrix seems similar with or without the quadrupole moments (see for instance [50]. However, for quantitative description and for (perhaps) solving the gate puzzle, careful investigation of different energy parameterizations are very much to be desired.

In an ideal computer simulation, efforts are made to reproduce the experimental set-up as accurately as possible. A ligand is placed in the heme pocket, bonded or restrained to remain at the binding site. It is kept that way for an equilibrium simulation in which water and protein are allowed to relax to an equilibrium state appropriate to the force field used. The structures generated at equilibrium of the bound state are sampled to initiate trajectories of ligand diffusion and recombination. In most of the conducted simulations the trajectories are purely diffusive ([24, 50–52], but in some cases they include detailed description of recombination events (re-formation of ligand – heme bond [48, 53]). Simulations on multiple electronic energy surfaces that describe the bound and unbound ligand-protein state provide a complete computational description of recombination measurements [54]. They make it possible to describe the photo-dissociation, energy relaxation, ligand diffusion, and re-binding in a single computational framework. This approach is most appropriate for direct comparison with flash photolysis and geminate recombination experiments. N [48] correct ordering of nitric-oxide recombination-rates of myoglobin mutants were obtained.

Recently however, the focus was on one component of the experiment, that of ligand diffusion between internal cavities. The availability of time resolved FTIR spectroscopy and X-ray crystallography experiments that watch the build up of ligand densities at different location in the protein matrix (figure 1) makes direct comparison with experiment possible.

A number of computational groups have taken on themselves the task of simulating the ligand diffusion in and out of myoglobin using straightforward tools of Molecular Dynamics without additional physical assumptions that enhance the computational efficiency. Of the studies mentioned above perhaps the most comprehensive is the investigation by Ruscio et al. [51] which, as described, should have put to rest many of the lingering questions about links between experiment and theory in this comprehensively studied system. Sixty-eight trajectories were conducted for ninety nanoseconds each providing ample evidence and statistics for ligand migration in the protein and escape. Moreover, since the simulated system is “small” (or the concentration of myoglobin in the simulated system is very high), entries events from solvent to protein were also observed. The time scale is in overall agreement with rates measured experimentally (about 100–200 nanoseconds) and the diffusion pathways are in remarkable agreement with the emerging consensus of ligand hopping between the Xenon sites [5]. Nevertheless, one qualitative question remains and with this question I wish to conclude the present Opinion.

Conclusion

In a beautiful series of experiments Scott et al. showed that the dominant escape pathway for a diatomic ligand from the protein matrix is the histidine gate [7] including an in depth examination of ligand movement in W29 mutant of myoglobin by time resolved X-ray crystallography [55]. This study was further confirmed by a series of mutations to tryptophan residues at critical points along the ligand diffusion pathway, a study that is summarized in a review [16]. These observations appear to contradict all simulations which have searched for escape pathways (rather than assuming them). A recent study in the laboratory of the author [52] observed two non-classical (not histidine) exits and the study by Ruscio et al [51] proposed nine gates (with different surface residues). The simulation results, of multiple roughly equivalent exit pathways, are not supported by current experiments [7, 16]. Kinetic analysis indicates that 70–80% of the ligands enter and exit from the distal histidine gate.