CONCLUSIONS AND IMPLICATIONS OF THE CROWDED CHARGE REDUCED MODEL

It seems clear that the reduced models channels as charged spheres in a small space are quite successful. These simple models using the simplest kind of physical chemistry deal with complex biology. It is also clear that this model is successful for two main reasons: (1) it calculates the energies that biology seems to use to produce selectivity in these channel types; (2) it calculates a self-organized binding site, with an induced fit between ions and side chains and vice versa. The approach of the physicist—“guess and check,” then add complexity—seems to have worked. The implications of this success seem significant. It provides an alternative path to the common approach using molecular dynamics, best shown in the huge literature on the selectivity of the potassium channel already cited. The common approach tries to compute everything, despite the enormous gaps in scales that make this so difficult (see Table I). Clearly, more resolution will be needed than we have used so far, as other channels and other properties of calcium and sodium channels are computed. All those features of the protein are there for a reason. The lysine of the DEKA channel does more than just contribute charge in all likelihood. But as we add resolution, it is important to preserve the features that make the low-resolution model successful.

It seems that higher resolution models must share the important features of the reduced model if they are to share this success. Higher resolution models can deal with calcium channels (for example) only if they compute energies with similar properties to those computed by the reduced model under the range of conditions that the calcium channel model succeeds in fitting data. Specifically, a higher resolution model—applied to the same system of spheres—should give results nearly the same as the reduced model in a range of concentrations of Ca2+ from 10−7to 1 M, and a wide range of Na+, K+, and so on, concentrations, and a range of divalent concentrations as well. Just as importantly, higher resolution models should be shown to change selectivity when the side chains are changed from EEEE to DEKA.

So far higher resolution models of molecular dynamics, using elaborate force fields to describe interatomic forces, have not yet been shown to reproduce the

results of simple models correctly. If high-resolution simulations do not deal with the issues known to be important in the laboratory, it is not clear how higher res- olution simulations can deal with real laboratory results. Higher resolution mod- els need to include specific concentrations in the bath as inputs. Higher resolu- tion models must estimate activities of individual ions with reasonable accuracy. Experimentalists know they must estimate activities reasonably accurately even to identify the type of channel they are studying. They know that most of the properties of proteins and channels depend on the concentration of ions. So far higher resolution models of molecular dynamics, using elaborate force fields to describe interatomic forces, have not yet been able to deal with a range of ionic concentrations in mixed solutions. They have not been able to calculate activi- ties over a range of concentrations or in mixtures, or even in pure solutions of divalents like CaCl2. Thus, at this stage we believe that high-resolution models of selectivity are not yet ready to be compared with low-resolution models, or with experimental data.

Despite all this discussion, it is obvious that higher resolution is needed than the crude reduced models that we have used so far. More details in the structure are certainly involved in some functions of the calcium and sodium channels we have dealt with. More details of the structure may be involved in the main selectivity or binding properties of other channels and proteins. Molecular dynamics with force fields is in fact a kind of reduced model, because the parameters of the model are determined (in large measure) from fits to macroscopic data. The advantage over the reduced models used here is that they include all atoms. The disadvantage is that they do not deal with ions very well in pure solutions, let alone in concentrated mixtures. Perhaps an important difficulty is the choice of macroscopic data. Perhaps when force fields are calibrated against measurements of the activity of ions they will do better.

Until higher resolution models are actually calibrated against simpler models, it will be difficult to compare the two classes of models, however. The tradition of the physical sciences assumes such calibrations are necessary in most cases. Certainly, engineers know that calibration is necessary if their devices and machines are to work or are to improve as complexity is added. Biological scientists perhaps need to learn here from our physical colleagues.

If the models cannot be compared, science as we know it cannot proceed, in my view. The scientific method requires us to be able to choose between models, at least in principle. If we cannot, because our computations are inadequate or our experiments are incomplete, we should do something else, until our technology advances to where we can do something useful. The scientific endeavor of “guess and check” can be viewed as a social process, that is justified if it discovers some- thing useful, or builds something that works. “Guess and check” cannot converge to a useful result (to use mathspeak), if check is impossible. If different models cannot be distinguished, checking them is impossible and science, as I define it,

does not work. The need for checking and the need for calibration are essential components of the scientific process I think.

The need for calibration of molecular dynamics thus seems self-evident to some of us [221, 222, 224]. The issues are not just the force fields, but the numerical difficulties themselves (which we shall describe later). Work actually comparing properties of ions in solution with ions in known physical systems is just beginning [116, 120, 122, 124–127, 130, 387, 500], also see Refs [224, 648, 651]. Extending this work to deal with experimental reality is a great challenge but certainly one that can be met as computational size and scientific wisdom increase, hopefully at comparable rates.