Different maths for dealing with the gating particle???

Different maths for dealing with the gating particle???

Postby Bill Connelly » Fri Feb 01, 2013 1:53 pm

I "understand" the concept that if we have the reaction
A <--> B
With the forward rate Kf and the backwards rate Kb, then we can say
dA/dt = -Kf . A + Kb . B

From that, I understand, when it comes to HH kinetics, we could say, in the DERIVATIVE block
n' = alpha(v)*(1-n) - beta(v)*n

However, I very rarely see NMODL coded in that form. It is normally in the form...
n' = (n_inf - n) / tau_n

which is calculated from
tau_n = 1 / (alpha(v) + beta(v))
n_inf = alpha(v) / (alpha(v) + beta(v))

This, I do not understand. I assume these later "alpha" and "beta" functions are not identical to the former ones, if the mechanism was to produce an identical current? How do I think about these alphas and betas... are they still foward and backward rate constants? Why do people not use the first form that seems more intuitive? It seems hard to believe it is for a computation advantage. Finally, and most importantly, how do I adjust said alpha and beta functions to adjust the kinetics in terms an electrophysiologist would understand, i.e. the V50, the activation kinetics etc?

For reference sake, here is the alpha and beta from the mechanism I am looking at

alpha = 0.032 * (15-v2) / ( exp((15-v2)/5) - 1)
beta = 0.5 * exp((10-v2)/40)
Bill Connelly
 
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Re: Different maths for dealing with the gating particle???

Postby ted » Sat Feb 02, 2013 11:58 am

Take conservation into account and you'll see the equivalence between the ODE and kinetic scheme formulations.
Start with
A <-> B with forward and backward rates a and b (I hate typing)
and impose the assumption that A+B = 1 (makes sense if A and B are the fractions of "gating particles" that are in the "closed" and "open" states, respectively)

dB/dt = aA - bB = a(1 - B) - bB = a - (a + b)B

My claim is that this is equivalent to
dB/dt = (Binf - B)/tauB = Binf/tauB - B/tauB

If x = y and x = z then y = z, so
a - (a + b)B = Binf/tauB - B/tauB

The first term on the left hand side is a (a constant), and the second term is the product of (a + b) (another constant) times B (a state variable).
The first term on the right hand side is Binf/taub (a constant), and the second term is B (a state variable) divided by tauB (a constant).

The equality
a - (a + b)B = Binf/tauB - B/tauB
demands that the constant terms on the right and left hand sides equal each other,
and that the state-variable-involved terms on the right and left hand sides equal each other.
In other words, we have these two new equations
a = Binf/tauB
and
(a + b)B = B/tauB
and we can solve these two new equations for tauB and Binf to get
tauB = 1/(a + b)
and
Binf = a tauB = a/(a + b)
ted
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Re: Different maths for dealing with the gating particle???

Postby Bill Connelly » Mon Feb 04, 2013 5:55 am

Very interesting...

So then is there any computation reason why someone would essentially insert a few more lines of code that convert alpha and beta into tau and inf, and then essentially convert it back later on? I assume it is for some conceptual reason?
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Location: Australian National University

Re: Different maths for dealing with the gating particle???

Postby ted » Mon Feb 04, 2013 11:06 am

Many voltage-gated channels use an ODE-based idiom similar to
Code: Select all
BREAKPOINT {
        SOLVE states METHOD cnexp
        gk = gkbar*n*n*n*n
        ik = gk*(v - ek)
}
INITIAL {
        rates(v)
        n = ninf
}
DERIVATIVE states { 
        rates(v)
        n' =  (ninf-n)/ntau
}
PROCEDURE rates(v(mV)) {
        LOCAL  alpha, beta
        alpha = something
        beta = somethingelse
        ntau = 1/(alpha + beta)
        ninf = alpha*ntau
}
instead of the kinetic scheme alternative. Probably the most common reason is the long chain of precedence starting with Hodgkin & Huxley. For NEURON users, practical justifications for using ODEs are that the cnexp method is fast and, if secondorder is set to 1 or 2, generates second order correct solutions (v error is proportional to dt^2). Kinetic schemes require the sparse method, which is slower than cnexp and produces first order correct solutions (v error proportional to dt)--see Integration methods for SOLVE statements in the Forum's Hot tips area. That said, some voltage- and/or ligand-gated channel models are expressed as Markov models--multiple states coupled by transitions with v- or concentration-dependent probabilities--especially in the pharmacology and channel biophysics literature. Those are most easily implemented with a KINETIC scheme, i.e. a family of "reactions". Yes, there are some published examples of code whose authors have laboriously translated a Markov model to a family of ODEs, but the resulting code is long and difficult to read, debug, and maintain; it might run a bit faster than if expressed as family of reactions, but programmer's time is generally far more valuable than computer time.
ted
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