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Anatomy of a model

We shall start with the Hodgkin-Huxley equations that describe the excitable process of a giant squid axon. Although much of what follows is our speculation, we suspect that our rationale for each equation is quite similar to theirs.

The equations that we are about to derive are based on the definitions of the current-voltage relationship for different circuit components. Combining circuit elements alters the total current-voltage relationship, and hence, the behavior of the circuit. Here we use $ V$ to be the potential difference across the circuit element, $ I$ to be the current, the amount of charge, $ q$, that flows per unit time through the element. $ V$, $ I$ and $ q$ are functions of both time, $ t$ and space, $ x$.

  1. Ohm's Law: $ I = g V$, where $ g$ is the conductance (the reciprocal of $ R$, resistance) and represents the proportionality constant relating current to the difference in potential across a resistor. This implies that the current through a resistor is linearly proportional to the difference in potential across the resistor (the relationship used to describe current through conducting membrane ion channels).
  2. Definition of Capacitance: $ q = C V$, where $ C$ is the capacitance of the circuit element and is the proportionality constant relating charge with potential. This implies that the charge stored within a capacitor is linearly proportional to the difference in potential across the capacitor.
  3. Current is the amount of charge that flows/unit time so taking derivatives of the above, we have:

    $\displaystyle I = \frac{\textrm{d}q}{\textrm{d}t} = C \frac{\textrm{d}V}{\textrm{d}t}.$    

Circuits are constructed by parallel or series combinations of resistors, capacitors and inductors, however, there are few biological analogs of inductors and we will ignore them in this context. The differential equations that describe the behavior of a circuit are derived by applying Kirchoff's conservation laws to the circuit:

  1. Kirchoff's Current Law: All of the current that flows into a node (an intersection of 2 or more circuit elements) must be equal to the amount of current that flows out of the node. For example, if a circuit element has one input and two outputs and one amp flows into it, then one amp must be distributed between the two outputs.
  2. Kirchoff's Voltage Law: The sum of the voltage differences measured around a loop of circuit elements must be zero.

Applying these principles to biological systems yields equations that can often characterize a surprisingly large amount of behavior. The fun of modeling is to identify the minimal model required to capture the behavior of a biological process.

In studies of the relationship between current passing across the membrane of a squid giant nerve axon, H-H observed two major currents in response to a step change in the transmembrane potential, an inward Na current that rapidly turned on (activated) and off (inactivated), and a slowly activating outward K current (delayed rectifier) as shown in figure 2.5.1. They incorporated a third current, a leakage current, in order to maintain a balance of current under rest conditions.

Figure: Computed ionic currents for squid giant axon.

Each current was characterized by Ohm's law, $ I = g V$, but because the ionic currents flowed according to different gradients2.1, the potential, $ V$, must be related to the reversal potential2.2, $ V_i$, where $ i$ is simply a label for the type of current. The total current is thus the sum of two components, the field component, $ g_{i}V$ and the ion gradient component, $ -g_{i}V_{i}$. Combining the two we can write Ohms law as

$\displaystyle I_{i} = g_{i} (V - V_{i}).$    

The effect of the gradient current can be seen in Figure 2.5.3 where the Na current goes from being negative to positive when the potential is a little over +40 mV. Thus, the reversal potential for Na is +40 mV. For K, the reversal potential is about -80 mV. The sign and the strength of the reversal potential is determined by the different gradients of ions.

H-H considered the membrane as an insulator surrounded on each side by a conductor (extracellular and intracellular fluid). Thus, the membrane acts as a capacitor where the amount of charge that can be stored on the insulating surface is $ q = C V$. Postulating that the membrane is composed of ion channels that control ion flow between the extracellular and intracellular fluids, the equivalent electrical circuit is the parallel combination of a capacitor, the membrane, and 3 conductances, Na, K and leakage. The current associated with each component of the circuit can be represented by the terms:

\begin{displaymath}\begin{array}{rl} C \frac{\textrm{d}V}{\textrm{d}t} & \textrm...
..._{\textrm{L}}) & \textrm{Leakage - channel current} \end{array}\end{displaymath}    

where $ b_{\textrm{Na}}$ and $ b_{\textrm{K}}$ are the gating terms and represent the fraction of ion channels that are open at a given time. Thus, if all of the Na channels are open, then $ b_{\textrm{Na}} = 1$ and you get full conductance for Na. However, if only half of the channels are open, the conductance is scaled by one half. From Kirchoff's current law, the sum of currents flowing into and out of a node (where circuit element are connected) is zero, we have

$\displaystyle C \frac{\textrm{d}V}{\textrm{d}t}$ $\displaystyle + g_{Na}b_{\textrm{Na}}(V - V_{Na}) + g_K b_{\textrm{K}}(V - V_K)$    
  $\displaystyle \quad + g_L(V - V_L) = 0$    

To extend the model to include propagation, unidirectional movement of ions from cell to cell, we have to add two additional sources of current in this balance, diffusive current into the node and diffusive current out of the node. The current into the node is $ \frac{V(x-\Delta x) - V(x)}{R \Delta x}$, where $ R$ is the internal resistance per unit length (the reciprocal of $ g$, conductance). It is easy to imagine that throughout the length of a cell that there would be all sorts of obstacles such as intracellular organs or proteins in the cytosol that would inhibit the free flow of ions through the cell. Since this hindrance is relatively uniform for the different types of ions, we only need on term to account for them. The current out of the node is $ \frac{V(x) - V(x + \Delta x)}{R \Delta x}$. Because we cannot manufacture charge, then the difference per unit length must equal the current through the membrane.

Figure: Flow of diffusive and ionic current within a nerve or cardiac cell. Ionic currents flow down a potential gradient along the radial axis of a nerve or muscle cell, and, as membrane ion channels open, ions can flow down transmembrane concentration and potential gradients. Typically Na and Ca ions flow into the cell while K ions flow out of the cell.

  $\displaystyle \frac{\frac{V(x-\Delta x) - V(x)}{R \Delta x} - \frac{V(x) - V(x + \Delta x)}{R \Delta x}}{\Delta x} =$    
  $\displaystyle \quad \quad C \frac{\textrm{d}V}{\textrm{d}t} + g_{\textrm{Na}}b_...
...trm{Na}}(V - V_{\textrm{Na}}) + g_\textrm{K} b_{\textrm{K}}(V - V_{\textrm{K}})$    
  $\displaystyle \quad \quad \quad + g_{\textrm{L}}(V - V_{\textrm{L}})$    

Taking the limit as $ \Delta x$ goes to zero then we have the standard nonlinear parabolic partial differential equation with the driving function composed of the individual ionic currents.

$\displaystyle \nabla^2V = \frac{\partial V}{\partial t} + \sum I_i$ (2.5.1)

where $ I_1 = I_{\textrm{Na}}$, $ I_2 = I_{\textrm{K}}$ and $ I_3 =

At this point, the two gating variables, $ b_{\textrm{Na}}$ and $ b_{\textrm{K}}$, were defined by words, but there was no formula to define their value. These gating parameters separate the H-H model from pure first principles, Ohm's law and conservation of charge. This is where Hodgkin and Huxley strayed from ordinary science to extraordinary science and this yielded a Nobel prize.

Hodgkin and Huxley's experiments revealed that when they switched the potential across the cell membrane from a polarized value (-60 mV) to a depolarized value, 0 mV, and they poisoned the K charge carriers so that they saw only Na current, the Na current decreased transiently and then returned to zero (or near zero) (the red trace in Figure 2.5.1). This feature probably led them to conjecture that there was some sort of dynamic gating process that controlled the flow of ions through the channel. For Na current, they initially suggested two gates, one for activation and one for inactivation. Similarly for potassium, they initially suggested a single activation gate (the green trace in Figure 2.5.1).

To test their model they must have plotted observed and expected currents. From these plots they would have observed that the first draft of the model did not fit the initial onset of the activation process for Na or K. For Na current, three activation gates resulted in a better fit. The result was that they defined $ b_{\textrm{Na}} = m^3h$, where $ m$ is the probability that an activation gate opens after the nerve is stimulated, thus $ m^3$ is the probability that three open, and $ h$ is the probability the inactivation gate slowly closes after the nerve is stimulated. Similarly for the potassium current, they found that four gates fit the onset of activation better than a single gate and thus, $ b_{\textrm{K}} = n^4$. Substituting terms, our final equation is:

$\displaystyle C\frac{\textrm{d}V}{\textrm{d}t} + g_{\textrm{Na}}m^3h(V - V_{\textrm{Na}}) + g_{\textrm{K}}n^4(V - V_{\textrm{K}})$    
$\displaystyle + g_{\textrm{L}}(V - V_{\textrm{L}}) = 0$    

and the results are shown in Figure 2.5.4 where we show the computed action potential and the three gating variables.

Figure: Comparison of a single and 3 activation gates for cardiac Na channels. Experimentally observed peak currents shown as +, single gate as red and 3 gates as blue

This curve fitting exercise is a wonderful example of paying attention to small details. The current voltage relationship of a Na channel is usually measured by holding the cell at some negative (-120 mV) potential and then testing the response with a short duration (5 ms) shift in potential to a test potential. The peaks of each response is plotted and then you think about what the I/V curve is trying to tell you. Shown in 2.5.3 are peak Na currents measured in cultured cardiac cells by Gus Grant (+) as a function of the test potential and fits to the I/V curve assuming one (red) and 3 (blue) activation gates. Note that for the initial activation between -60 mV and -45 mV, the red (single) curve overestimates the current while the blue curve is right on the money. Similarly, the single activation gate overestimates the peak of the curve. Such consistent overestimation is typically not due to noise, but would suggest that something is missing from the model. In the H-H days of desk-top calculators, many (probably including me) would have been happy to get general agreement between observed currents and model currents as shown by the red line, but not Hodgkin and Huxley. They must have realized that there was something not quite right and redid their analysis for a 3 gate process for the Na channel and a 4 gate processes for the K channel.

Figure: Computed squid action potential and gating variables.

The first hint that the theory developed by H-H was correct came about 30 years later when the Na channel was cloned and sequenced by Noma and colleagues in Japan. They observed 4 subunits, each with 6 membrane spanning components and a perfect arrangement for a helical gate. 50 years later, Ray MacKinnon and colleagues managed to crystallize K channels and found four paddles that acted as voltages sensors for the gating process.

next up previous index
Next: Model Approximations and Assumptions Up: How to create a Previous: Ordinary Differential Equations   Index

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Frank Starmer 2004-05-19