Biophysical Substrate  Second Messenger Cascades

Multiplication and Second-Messenger Cascades

``Chemistry allows multiplication.''

We have shown that the rate of channel transitions (or the occupancy of transition states) is proportional in equilibrium to $\bigl[ {\displaystyle{d \over dV}} m_\infty(V) \bigr]^\gamma$,where $\gamma$ is some exponent such as $1/2$.

  Since the products of second-messenger cascades depend on the law of mass action, the rate of production or concentration of the effector, or end product of the cascade, can be proportional to $m_\infty(V)$ or $\bigl[ {\displaystyle{d \over dV}} m_\infty(V) \bigr]^\gamma$raised to some power. For instance, a G-protein mediated second messenger cascade can take

$$\qquad \qquad {\Bigr[ {d \over dV} m_{\infty}(V) \Bigr]}^{ 1... ...{d \over dV} m_{\infty}(V), \qquad \text{where $n$ is integer.}$$ In particular, we are interested in the case $n = 2$, since this corresponds to the equilibrium model of a voltage-dependent ion channel with two states. Suppose the activated form of the G-protein, which is produced at a rate proportional to $[{\text{O}}^\ast ]$, or the ion channel transition frequency, gives rise to two products, $A$ and $B$. Suppose that the effector $E$,which could be a protein kinase, transcriptional regulatory protein, depends on $A$ and $B$ as follows: $$A + B + E \longrightarrow E^\ast .$$ In this case, the rate of $E^\ast$ production is proportional to the product $[A][B]$ of the concentrations of $[A]$ and $[B]$. Assuming that $[A]$ and $[B]$ quickly reach equilibrium values, the rate of effector activation will be $$\begin{align*}E^\ast(t) & \sim [A][B] \sim f_O^2\ & = {d\over dV} m_\infty(V).\end{align*}$$ (Here we assume that $E^\ast(t)$ is not inactivated, recycled, or degraded quickly and that $E$ is present in abundant quantity.)

If each voltage-gated channel is associated with its own G-protein-mediated mechanism for self-regulation, then the functions $m_\infty(V)$ and $\bigl[ {\displaystyle{d \over dV}} m_\infty(V) \bigr]^\gamma$for different voltage-gated ion channel types form a set of computational primitives. Using these primitives,  a neuron could maximize the information about stimuli in its firing rate .

3/7/1998