SEGUNDO MOMENTO DE FORMACIÓN (Ver anexo F)

Guess a model; check it; fix it, and add more if needed. One should start by assuming that the usual models and methods of physics and engineering can deal with biological complexity. The usual procedures of physics and engineering can then be employed to understand and control channels. If those procedures prove inadequate, new principles can be introduced, special to biology, if they are specific and quantitative, and testable.

We follow standard procedures of physical analysis here as we use reduced physical models and try to understand the selectivity of calcium and sodium chan- nels. We guess what physics may be involved. We derive as carefully as we can the consequences of that guess. We check the consequences, and modify the model as needed to improve the guess. “Guess and check” is the name of the game.

V. REDUCED MODELS OF CALCIUM AND SODIUM CHANNELS

Reduced models of channels are built in this tradition of guess and check. The hope is that these models capture the essential physics used by biology to create the selectivity important for biological function. The models are simple enough so that the physics they contain can be calculated with some accuracy. They are justified by their fit to data and by the robustness of their results: methods from MSA [182], to SPM [184], to Monte Carlo (MC) [378, 379], to DFT [380], and PNP-DFT [381] give essentially similar results, often quantitatively [182, 378, 379, 382] as well.

We concentrate on reduced models of calcium and sodium channels because they have been quite successful in dealing with the selectivity of these channels as measured over a wide range of conditions. These models represent the side chains of proteins as spheres of charge that occupy volume and interact with mobile ions (K+, Na+, Ca2+_{, and Cl}−_{) through volume exclusion and electrostatics much} as the mobile ions interact with each other. The ions and “side chains” mingle together in the selectivity filter of the channel, typically a region 10 ˚A long and 6 ˚A in diameter to which the side chains are confined. The solvent and protein are represented implicitly as dielectrics.

We hope that the principles of a general approach to channel permeation and selectivity emerge from this specific analysis, along with some general physics that may be present in most problems of channel and protein function. The general approach assumes that understanding of selectivity requires measurements in a wide range of solutions of different concentration and types of ions. Computations

of a “free energy of binding” in a single solution are not helpful for two reasons. Properties in a single solution are too easy to explain. It is difficult then to separate one model of selectivity from another. Second, it is clear that the “free energy of binding” is not constant, but depends on ionic conditions. The ions in the baths and channel are not ideal. Everything interacts with everything. The free energy of binding depends on all concentrations. Thus, calculations done under only one set of conditions are not very useful, for our purposes. They do not permit comparison with a range of data; they are vague enough that calculations with different models cannot be compared.

This reduced model of selectivity is in the long traditional of primitive im- plicit solvent models of ionic solutions. Models of bulk solutions using implicit solvents have a long and successful history in physical chemistry [116–119, 124, 126, 128–130, 383–391], and have been particularly investigated and compared with experiment in Turq’s group [384, 385, 392–399]. Implicit solvent models of proteins are also widely used in the study of protein function. Indeed, the literature of implicit solvent models of proteins is too large to review [2–4, 329, 355–359, 375, 400–431].

The lack of detail in implicit solvent models is primitive, as the name implies. The treatment of polarization as a dielectric is actually embarrassing to those of us who are aware of the complex dielectric properties of ionic solutions [432–436] and electrochemical systems in general [57, 178, 201, 260, 432, 434, 437–440]. I spent many years making impedance measurements of the complex dielectric properties of biological systems to determine their electrical structure [441–446] and so have measured the dielectric properties of cells, tissues, or ionic solutions that cannot be described by dispersion, or a single dielectric constant.

The dielectric “constant” of ionic solutions, in particular, is nothing like a con- stant. It varies from 80 to 2 in the time range of atomic motions relevant to molecular and Brownian dynamics (i.e., from 10−13s to 10−7s). The likelihood of nonlin- ear field dependence in the region close to an ion (particularly a multivalent ion) cannot be denied. Indeed, electron orbital delocalization may occur in some cases, and then solvation has some of the characteristics of a classical chemical reaction involving (partial) covalent bond formation. The fact is, however, that so far the most successful treatments of ionic solutions are primitive despite the impressive progress of a number of laboratories [116, 121, 124, 130, 383, 387]. Only the primitive model has allowed calculations of the fundamental properties of ionic solutions, namely their free energy per mole or chemical potential. Primitive mod- els are a good place to start. They are also surprisingly successful. Perhaps the most important properties of ionic solutions depend mostly on integrals of the dielectric properties over all frequencies because of the Kramers Kronig relations and this integral property is captured by implicit solvent models well enough.

The fundamental property of any ionic solution is its free energy per mole, its activity, or electrochemical potential, all nearly the same thing, differently

normalized, written in logarithmic, exponential, or linear scales [99, 118, 119, 126, 128, 129, 217, 390, 447, 448]. Almost all solutions have excess chemical potential, activity coefficients, or osmotic coefficients not equal to unity because few solutions are ideal.

The central fact of electrochemistry is that the excess chemical potential of an ion is not zero [120, 125]. The excess chemical potential in fact varies as the square root of its concentration (speaking loosely for 1-1 electrolytes like NaCl) and not linearly [119, 126, 390, 449]. Ions are not independent in ionic solutions.

Ions are not independent in ionic solids, where we take for granted the fact that there are exactly equal numbers of Na+ and Cl− ions (or we would be electro- cuted each time we salt our food [450]), and ions are not independent in solution [449] because of the screening [307] reflected in the fundamental sum rules [308, 309] describing ionic fluids. The requirement of electroneutrality in bulk solutions guarantees that ions in solutions have highly correlated behavior not found in ideal infinitely dilute gases of point particles without charge.

Ionic solutions are not ideal. Their extensive properties are not proportional to number density. The “independence principle” that Hodgkin and Huxley [160, 167–170, 211] used so brilliantly to understand the properties of nerve membranes does not apply to bulk solutions. The independence principle correctly describes the movement of different species of ions through different (and independent) protein channels in a membrane, if they are perfectly selective to those species. The independence principle describes almost nothing else.