Structures of proteins and ions change with conditions so that measurements of them would have to be made under the conditions in which they function if they are to show the structures that perform the function. This is a difficult requirement, more or less incompatible with X-ray crystallography as we know it. Crystal forma- tion requires special conditions, with special salts, detergents, and other conditions. The free energy of ions near the protein active sites are unknown in crystals. The number density and free energy of ions near the protein active sites are usually unknown in crystals. It is not reasonable to expect structures that are sensitive to experimental condition in simulations to lose that sensitivity in the laboratory even when they are in crystals at 100K.
None of this is to deny the enormous importance of structure for what it is: the average distribution of ions under special conditions. Models must certainly be compatible with those structures when the models are computed under the conditions in which the structures are measured. But structures measured in one (unknown) ionic condition cannot show how the structure changes under other conditions, let alone what the energetics and forces are in either case.
A further point needs making, judging from experience with students and col- leagues, even if it seems obvious. Most measurements of structure are made at 100K, where disorder is much less than at biological temperatures. The entropic contribution to free energy is proportional to the absolute temperature: it is roughly speakingTS. The difference in the entropic contribution at 100 and 300K is sub- stantial [552] and would have enormous effects in any calculation of rates that depend on the exponent of the free energy (see Ref. [553] and p. 1991 of Ref. [182] and Fig. 11 of Ref. [176]). Of course, the structures observed at 100K are not artifacts. Of course, many of the structures are visible at 300K. Of course, cold temperatures are used to minimize experimental artifacts (like crystal damage). Of course, it is also true that ions found in specific locations at 100K cannot be assumed to be there at 300K [553]. Ions found at specific locations at 100K may be there simply because the entropy at that temperature is too low to move them to where they normally reside and function. They may be “salted out” by the cold, to use some old fashioned language.
In my view, detailed physical analysis is required along with structural analysis, if selectivity and protein function are to be understood. Computations in atomic
detail starting with the dynamics of all atoms may someday provide such physical analysis, but they can only do that after the computations are calibrated to be sure they estimate macroscopic variables correctly. We know that macroscopic “thermodynamic” variables describe many biological functions essential to life. Concentrations of ions and average electrical potentials determine whether nerves conduct, muscles contract, and patients live. Computations in atomic detail must be able to compute these variables if they are to deal with the biological functions they describe.
Perhaps it is best to view the functionally relevant “structure of a protein,” as well as the structure of forces in a protein, as the computed consequence of a model, constrained by conventional crystallographic and functional data.
VIII. INVERSE METHODS AND SELECTIVITY MODELS
Determining the mechanism of selectivity is an example of “reverse engineering.” We wish to determine what is inside the black box of the selectivity filter from measurements taken outside. Inverse problems of this sort are notoriously difficult, and involve a set of mathematical problems [525, 526, 554] quite different from forward problems. In general, the central issue in reverse engineering or inverse problems is doing the right measurement. Inverse problems are usually ill-posed, without enough data present to give unique solutions, and often show extreme sensitivity to error. When a desired result (e.g., the energetics of calcium ions) is insensitive to the measurement, the measurement is not worth doing. When the desired result is too sensitive to the measurement, the measurement cannot be made well enough (with little enough systematic and random error) to give reliable estimates of the parameters of interest. It is the goal of the experimental scientist to make measurements of variables that allow the appropriate sensitivity, so the parameters of interest can be determined.
Despite their difficulties, inverse problems can be solved and are done so rou- tinely in industry [555], where they are an important tool in process design. The key is always to have enough data of the right kind. The inverse problem for the distribution of charge within a channel has been solved with these industrial meth- ods [321] using a PNP–DFT theory applied to the reduced model described here. Noad hocassumptions were needed. Calculations done with added noise and sys- tematic error were quite robust because of the large amount of data available from a range of ionic conditions, and the high signal-to-noise ratio of the measurements of current–voltage relations. In fact, the numerical problem was that of (far) too much data (for computers at that time) and not too little! Thus, one can have con- fidence that reduced models can be built that can be tested experimentally. This is one of the few problems in channel biology or perhaps molecular biology in which a formal mathematical treatment of the inverse problem has been made.
The key to solving the inverse problem in channels was to have a large number of measurements in many different solutions of different concentrations of different types of ions. Reduced models allow calculations of channel properties under this range of conditions. So far, high-resolution simulations have not been done in a range of concentrations. Indeed, many are done with an ill-defined concentration of ions with uncalibrated free energy. (That is to say, the activity of the ions is not known.) Reduced models are needed (so far) if calculations are to be done under the wide range of conditions needed to solve inverse problems.
The need to solve the inverse problem should not be viewed as a mathematical nicety. If the inverse problem cannot be solved, different investigators will not be able to distinguish their different models of how selectivity occurs. A search of the literature concerning the selectivity of K+ channels will show an enormous diversity of explanations, sometimes with more than one explanation per scientist or research group. I only cite a fraction of the enormous literature here [42, 171, 172, 174, 185–189, 219, 288, 292, 349, 354, 419, 485, 533, 556–622]. Because these explanations do not actually deal with experimental data over a range of concentrations of different types of ions, they cannot be told apart (in my view). In my view, calculations producing results that cannot be distinguished have limited utility. Reduced models are needed so far to do calculations in a large enough set of conditions that one can tell one explanation from another.
I argue again that reduced models have a general role in biology. Without reduced models, discussion is ill-posed. Different models cannot be distinguished. One cannot tell one idea from another.
I argue that biology can be analyzed by a series of reduced models, each with a complexity appropriate for the biological question be asked, each appropriate for the scale on which evolution has built that biological function. I argue in fact that the range of scales involved in biology forces us to use a sequence of reduced models. Inverse methods can help us choose these models intelligently. Variational methods allow us to compound models on different scales into one overall theory, at least in principle. Both methods are discussed later in this paper in a general context.
Practical scientists are not impressed by the theory of inverse problems, under- standably enough, since successful biologists are often people who wish to test all ideas themselves without the help of abstractions of mathematics or abstracted mathematicians. So now we move on to the specific case of calcium channels. Practical scientists can judge the utility of reduced models in everyday experimen- tal work. Calcium channels can actually be built using the design principles of these models [509, 510, 512].