All Atom Simulation of the Nerve Signal: the Impossible Dream


The recent Nobel Prizes for molecular dynamics have recognized the enormous growth and importance of the all atom view of biological molecules and their dynamics. All atom calculations are limited, however. All atom simulations of some biological functions are impossible, for all intents and purposes. This limitation does not minimize the objective importance of feasible calculations with atomic resolution nor the subjective (motivational) importance of unreachable goals. 

The nerve signal called the action potential carries information in more or less digital form over long distances (say 0.001 m to 10 m) in nerve cells, mostly nerve axons. The action potential is driven  by currents through two types of channel proteins, one that catalyzes the flow of sodium ions and the other that catalyzes the flow of potassium ions, much more than anything else. These channels open in response to changes in the electrical potential across the voltage sensors of the channels. The gradients of concentration, electrical, and electrochemical potential for sodium drive sodium ions (and electrical current) inward through the sodiumchannel when it opens. The gradients of concentration, electrical and electrochemical potential for potassium are different. They drive potassium ions outwards when the potassium channel opens.

Currents carried across the membrane by these channels spread down the length of axons as current flows down transmission lines according to equations worked out some time ago by Kelvin to describe current flow in telegraph cables under the ocean. Those equations show that the potential at one location depends on the current flow from other locations millimeters to centimeters distant. The potential at one location depends on the potential at other locations. Some 10^n ions interact this way, in a 0.5 mm diameter axon, to pick an extreme case in which the wave length of the action potential is XX cm, when propagating at a velocity of XX m/sec, with duration N msec.

It seems clear that a computation of the interactions of 10^n ions is not possible, even if water is neglected. The action potential cannot be computed with atomic resolution.

Of course, there is no need to compute an action potential with atomic resolution. The action potential is a macroscopic phenomenon and the multiscale description of the action potential is known, thanks to the work of Hodgkin, Huxley, and Cole recognized in Nobel Prizes to some of them, long ago. The atomic description of propagation is not needed. The atomic scale description of individual channels is nearly enough, because the interaction of channels is described accurately by an intermediate scale equation, the transmission line equation.

The voltage clamp technique of Cole and Hodgkin allows measurement of the properties of ensembles of noninteracting channels. That is why the technique was invented. Single channel recording allows measurement of current through one channel protein at a time using the patch clamp method (among others) recognized in the Nobel Prize work of Neher and Sakmann. The voltage and time dependence of ensembles of noninteracting channels is simple enough that the action potential can be reconstructed from measurements of channels in the voltage clamp and (nearly) from measurements of single channels themselves.

Calculations of current through single channels then replace calculations of the action potential itself. The multiscale equation of transmission lines is all that is needed to go from single protein molecules to the macroscopic biological funtion. Atomic scale simulations are needed then of the ions flowing through single channels, not of the multitude of ions involved in the propagration of the nerve signal. 

The challenge of calculating ions through single channel proteins is itself a multi-scale problem, however, and atomic resolution will be difficult to achieve. The reasons are obvious if one starts from the properties of these channels known from decades of experimentation. 

The properties of individual sodium and potassium channels depend sensitively on the solutions around them. These solutions are invariably ionic mixtures (made mostly of Na, K, Ca, and Cl ions) derived from roughly half molar seawater. Seawater is highly concentrated compared to the dilute solutions in which  solutions of either NaCl or KCl  have ideal properties. Most channels change properties dramatically if the concentration of calcium ions is changed, particuarly on the inside of the channel. The concentration of calcium inside channels is typically 0.0000001 M.

Computations in atomic detail of ionic mixtures containing calcium ions poses certain challenges for all atom simulations. Calibration against the main physical properties of these mixtures (the free energy per mole, i.e., the electrochemical potential) is needed, and the calibration must be accurate given the sensitivity found experimentally to details of composition. All atom calculation of trace concentrations of intracellualr calcium ions is a particular challenge since ~55 moles of water must be calculated for each 1e-7 moles of calcium ion in a 1e-7 molar calcium solution.

Time scales of action potentials are milliseconds (nerve) to nearly seconds (cardiac) and so all atom simulations must be computed on that time scale. Since the time step of all atom simulations is typically 1e-15 sec, differential equations must be integrated for something like 1e13 steps, producing certain difficulties of reproducibility (e.g.,, unlimited dependence of individual trajectories on initial conditions; chaotic behavior of ensembles of trajectories with serious sampling errors) and accuracy in numerical procedures. 

Flows link single channels to the action potential. Calculations of flow from one ionic mixture to another (at different electrical, chemical, and electrochemical potentials) are needed in an all atom simulation, for that reason. Equilibrium analysis (with a single solution on both sides of a channel) will not do. It would be well to show that the nonequilibrium simulations actually reproduce the nonequlibrium behavior of simpler related systems, like diodes and 'triodes' (bipolar and field effect transistors) as calibrated simulations ion flow through solutions or channels are performed.

All atom simulations of flows through ion channels may not be necessary. The daunting challenges listed above may not have to be faced in full, all at once, if a multiscale analysis is possible. Certainly, atomic detail will be crucial in some parts of that analysis. The magnificent cathedrals of channel structure recognized in the Nobel Prize to Rod Mackinnon have architectural detail that controls biological function. Changes in a handful of atoms is enough to dramatically change  the properties of single channels. But full atomic detail may not be needed to account for the driving forces (i.e., electrochemical potentails) of all of the ingredients of ionic mixtuers surrounding channels.

All atom simulations of ion channels remain an admirable dream, whether impossible or not, a dream as long as that dream is sought in calibrated simulations that actually reproduce the experimental properties of ionic mixtures, flows in mixtures, ion channels, and flows through channels as measured in the laboratory. The shortcuts of multiscale analysis are well focused (and motivated) by the problems of all atom simulations, in my view. Without calibration, however, the all atom dream can easily become a multi-faceted nightmare, in my view motivating enormous calculations of limited use in dealing with biological reality.