First order differential equation for transmembrane diffusion I want to model transmembrane diffusion using a first order reversible differential equation. The diffusion rate is known from literature. There are two volumes $V_{\mathrm{intra}}$ and $V_{\mathrm{extra}}$. The former is $10^{12}$ times smaller than the latter. $c_{\mathrm{intra0}}=0~\mathrm{mol~L^{-1}}$, and $c_{\mathrm{extra0}}=x~\mathrm{mol~L^{-1}}$, with $x&gt;0$.

Thus far I am using this:
$$\frac{\mathrm{d}c_{\mathrm{intra}}}{\mathrm{d}t}=k_{\mathrm{diff}}\cdot c_{\mathrm{extra}} - k_{\mathrm{diff}} \cdot c_{\mathrm{intra}}$$

However, this does not account for the fact, that $V_{\mathrm{extra}}$ is much larger than $V_{\mathrm{intra}}$. I expect $c_{\mathrm{extra}}$ to be virtually constant. So far the the simulation results in the equilibrium being at $c_{\mathrm{extra0}}/2$.

Can someone suggest a method to account for the volume discrepancy?



Edit:

From the outlined scenario above, I expect in REALITY, my $c_{\mathrm{extra}}$ to not noticeably change from its initial value due to the large difference between $V_{\mathrm{intra}}$ and $V_{\mathrm{extra}}$.

However, in the simulation I don't want to set $c_{\mathrm{extra}}$ as constant, because I want to apply a 'extracellular' decay rate later on. Basically, I want to be able to modulate the extracellular concentration over time and see how the concentration within the cell changes.

This is why I want to account for the volume. Maybe, one needs an entirely different approach for that?
The solution to your differential equation, subject to the prescribed initial conditions is $$c_{intra}=c_{extra0}(1-e^{-k_{diff}t})\tag{1}$$ under the constraint that $c_{extra}$ does not change from its initial value of $c_{extra0}$.  The true material balance constraint is $$V_{intra}c_{intra}+V_{extra}c_{extra}=V_{extra}c_{extra0}$$
which would yield a result virtually indistinguishable from Eqn. 1.

CASE WHERE $c_{extra}$ IS NOT CONSTANT

OK.  Here is the solution for the case where $c_{extra}$ is a function of time $c_{extra}(t)$, and $c_{intra}=c_{intra}(0)$ at time t = 0:

$$c_{intra}(t)=c_{intra}(0)e^{-k_{diff}t}+k_{diff}\int_0^t{e^{-k_{diff}\tau}c_{extra}(t-\tau)d\tau}\tag{2}$$
where $\tau$ is a dummy variable of integration and where $c_{extra}(t-\tau)$ represents the value of the concentration $c_{extra}$ at time $t-\tau$.  This solution reduces to the previous result if $c_{intra}(0)=0$ and $c_{extra}$ is constant.  

So, to find the value of $c_{intra}$ at time t, you need to use Eqn. 2 and perform the indicated integration (usually numerically).