mcell/docs/suggested_3way_rxn.lyx

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\begin_body

\begin_layout Subsection*
Reactions with More than Two Components
\end_layout

\begin_layout Subsubsection*
Three-way reactions in MCell
\end_layout

\begin_layout Standard
The rate of reaction of a molecule that can engage in a three-way reaction
 with reactants 
\begin_inset Formula $I$
\end_inset

 and 
\begin_inset Formula $J$
\end_inset

 at concentrations 
\begin_inset Formula $\rho_{I}$
\end_inset

 and 
\begin_inset Formula $\rho_{J}$
\end_inset

 is 
\begin_inset Formula $\kappa\rho_{I}\rho_{J}$
\end_inset

.
 Suppose that a single molecule moves a distance 
\begin_inset Formula $R$
\end_inset

 while sweeping out an interaction area 
\begin_inset Formula $A$
\end_inset

.
 Then the expected number of hits, assuming that the concentration of 
\begin_inset Formula $I$
\end_inset

 and 
\begin_inset Formula $J$
\end_inset

 is low, is 
\begin_inset Formula \[
n_{\mathrm{{hits}}}=R\, A\,\rho_{I}\cdot R\, A\,\rho_{J}\]

\end_inset

 Thus, the expected number of hits for a molecule with a diffusion length
 constant of 
\begin_inset Formula $\lambda$
\end_inset

 is 
\begin_inset Formula \[
n_{\mathrm{{hits}}}=\int_{0}^{\infty}\rho_{I}\rho_{J}A^{2}R^{2}\frac{4\pi R^{2}}{\pi^{3/2}\lambda^{3}}e^{-R^{2}/\lambda^{2}}dR=\frac{3}{2}\rho_{I}\rho_{J}A^{2}\lambda^{2}\]

\end_inset

 If we let 
\begin_inset Formula $p$
\end_inset

 be the probability of reaction, then
\begin_inset Formula \[
\kappa\rho_{I}\rho_{J}\Delta t=p\cdot n=p\cdot\frac{3}{2}\rho_{I}\rho_{J}A^{2}\lambda^{2}\]

\end_inset

 Solving for 
\begin_inset Formula $p$
\end_inset

 gives 
\begin_inset Formula \[
p=\frac{\kappa}{6D\, A^{2}}\]

\end_inset

 assuming that 
\begin_inset Formula $\Delta t$
\end_inset

 is the time step for the moving molecule.
 If we let all three reactants move and react---let us number them 1, 2,
 and 3---then we matching the total rate gives 
\begin_inset Formula \[
\kappa\rho_{1}\rho_{2}\rho_{3}\Delta t=\frac{3}{2}\rho_{1}\rho_{2}\rho_{3}A^{2}\left(p_{1}\frac{\Delta t}{\Delta t_{1}}\lambda_{1}^{2}+p_{2}\frac{\Delta t}{\Delta t_{2}}\lambda_{2}^{2}+p_{3}\frac{\Delta t}{\Delta t_{3}}\lambda_{3}^{2}\right)\]

\end_inset

 where in general the individual molecules may move with custom timesteps
 
\begin_inset Formula $\Delta t_{i}$
\end_inset

.
 We let 
\begin_inset Formula $p_{1}=p_{2}=p_{3}=p$
\end_inset

 to give 
\begin_inset Formula \[
p=\frac{\kappa}{6\left(D_{1}+D_{2}+D_{3}\right)A^{2}}\]

\end_inset

 This solution also works for the cases where some of the reactants can't
 move (as 
\begin_inset Formula $D_{i}$
\end_inset

 will be zero and will drop out of the equation).
\end_layout

\begin_layout Standard
Now suppose that the reaction takes place near a surface such that for a
 fraction 
\begin_inset Formula $a$
\end_inset

 of the distance, the molecule sweeps out 
\begin_inset Formula $A^{\star}<A$
\end_inset

 of area instead of 
\begin_inset Formula $A$
\end_inset

.
 The expected number of hits is then 
\begin_inset Formula \[
n_{\mathrm{{hits}}}^{\star}=R\,\rho_{I}\,\left((1-a)A+aA^{\star}\right)\, R\,\rho_{J}\,\left((1-a)A+aA^{\star}\right)\]

\end_inset

 which we can rewrite as 
\begin_inset Formula \[
n_{\mathrm{{hits}}}^{\star}=n_{\mathrm{{hits}}}\left((1-a)^{2}+2a(1-a)\frac{A^{\star}}{A}+a^{2}\left(\frac{A^{\star}}{A}\right)^{2}\right)\]

\end_inset

 where the first term occurs when both hits are in the unconstrained space,
 the second when one target molecule is in the unconstrained space and one
 is in the constrained space, and the third when both targets are in the
 constrained space.
 If we multiply the probability of reaction by the inverse of the fractional
 areas for each target, i.e., by 
\begin_inset Formula $A/A^{\star}$
\end_inset

 if one target is in the constrained space and 
\begin_inset Formula $\left(A/A^{\star}\right)^{2}$
\end_inset

 when both are in the constrained space, we then find that the total rate
 of reaction is 
\begin_inset Formula \[
pn_{\mathrm{{hits}}}\left((1-a)^{2}\cdot1\cdot1+2a(1-a)\frac{A^{\star}}{A}\cdot1\cdot\frac{A}{A^{\star}}+a^{2}\left(\frac{A^{\star}}{A}\right)^{2}\cdot\frac{A}{A^{\star}}\cdot\frac{A}{A^{\star}}\right)=pn_{\mathrm{{hits}}}\]

\end_inset

 That is, the reaction rate is unchanged, which is exactly what we want.
 Since 
\begin_inset Formula $a$
\end_inset

 is arbitrary, we can make 
\begin_inset Formula $a$
\end_inset

 differentially small and thus the result holds for arbitrary restrictions
 of the swept area.
\end_layout

\begin_layout Subsubsection*
3-way reactions plus a surface in MCell
\end_layout

\begin_layout Standard
If we have three reactants but one of them is on a surface, we can calculate
 the probability of the moving molecule both striking the surface and hitting
 the other reactants.
 If the molecule is a distance 
\begin_inset Formula $h$
\end_inset

 above the surface, then the probability of hitting the surface is 
\begin_inset Formula \[
p_{\mathrm{surfhit}}=\frac{1}{2}\int_{h}^{\infty}dr_{z}\cdot\frac{1}{\pi^{1/2}\lambda}e^{-r_{z}^{2}/\lambda^{2}}\]

\end_inset

 But the molecule also diffuses in the 
\begin_inset Formula $xy$
\end_inset

 plane, and this determines the length of the collision cylinder.
 If the diffusion distance is 
\begin_inset Formula $r_{xy}$
\end_inset

 in that plane, then the total diffusion length is 
\begin_inset Formula $r=\sqrt{r_{xy}^{2}+r_{z}^{2}}$
\end_inset

 and the swept volume is 
\begin_inset Formula $A\sqrt{r_{xy}^{2}+r_{z}^{2}}$
\end_inset

.
 For a given 
\begin_inset Formula $r_{z}$
\end_inset

, then, the probability of hitting another free molecule is
\begin_inset Formula \[
p_{\mathrm{freehit}}=\int_{0}^{\infty}dr_{xy}\cdot\frac{2\pi r_{xy}}{\pi\lambda^{2}}e^{-r_{xy}^{2}/\lambda^{2}}\cdot\left(A\sqrt{r_{xy}^{2}+r_{z}^{2}}\right)\cdot\rho_{1}\]

\end_inset

where 
\begin_inset Formula $\rho_{1}$
\end_inset

 is the density of the other free molecule.
 Thus, the probability of hitting the surface from distance 
\begin_inset Formula $h$
\end_inset

 and also hitting a diffusing partner is 
\begin_inset Formula \[
\begin{array}{rcl}
p_{\mathrm{hit}}(h) & = & \frac{1}{2}\int_{h}^{\infty}dr_{z}\frac{1}{\pi^{1/2}\lambda}e^{-r_{z}^{2}/\lambda^{2}}\int_{0}^{\infty}dr_{xy}\cdot\frac{2\rho_{1}A}{\lambda^{2}}\, r_{xy}\sqrt{r_{xy}^{2}+r_{z}^{2}}\cdot e^{-r_{xy}^{2}/\lambda^{2}}\\
 & = & \frac{\rho_{1}A}{\pi^{1/2}\lambda^{3}}\int_{h}^{\infty}dr_{z}\cdot e^{-r_{z}^{2}/\lambda^{2}}\int_{0}^{\infty}dr_{xy}\cdot r_{xy}\sqrt{r_{xy}^{2}+r_{z}^{2}}e^{-r_{xy}^{2}/\lambda^{2}}\end{array}\]

\end_inset


\end_layout

\begin_layout Standard
The inner integral evaluates to 
\begin_inset Formula $\frac{\lambda^{2}}{2}r_{z}+\frac{\pi^{1/2}\lambda^{3}}{4}e^{r_{z}^{2}/\lambda^{2}}\mathrm{erfc(\frac{r_{z}}{\lambda})}$
\end_inset

, giving
\end_layout

\begin_layout Standard
\begin_inset Formula \[
p_{\mathrm{hit}}(h)=\frac{\rho_{1}A}{2\pi^{1/2}\lambda}\int_{h}^{\infty}dr_{z}\cdot\left(r_{z}e^{-r_{z}^{2}/\lambda^{2}}+\frac{\pi^{1/2}\lambda}{2}\mathrm{erfc}\left(\frac{r_{z}}{\lambda}\right)\right)\]

\end_inset

 The outer integral works out to 
\begin_inset Formula $\lambda^{2}e^{-h^{2}/\lambda^{2}}-\frac{\lambda h\pi^{1/2}}{2}\mathrm{erfc}(\frac{h}{\lambda})$
\end_inset

: 
\begin_inset Formula \[
p_{\mathrm{hit}}(h)=\frac{\rho_{1}A}{2\pi^{1/2}}\left(\lambda e^{-h^{2}/\lambda^{2}}-\frac{h\pi^{1/2}}{2}\mathrm{erfc}\left(\frac{h}{\lambda}\right)\right)\]

\end_inset

 which we can integrate over the entire column above a surface molecule
 of area 
\begin_inset Formula $B$
\end_inset

 to get the expected number of hits:
\begin_inset Formula \[
n_{\mathrm{hit}}=\int_{0}^{\infty}dh\cdot B\cdot\rho_{2}\cdot p_{\mathrm{hit}}(h)=\frac{\rho_{1}A\rho_{2}B}{2\pi^{1/2}}\cdot\frac{3\lambda^{2}\pi^{1/2}}{8}=\frac{3}{16}\lambda^{2}\rho_{1}\rho_{2}AB\]

\end_inset

 Since 
\begin_inset Formula $n_{\mathrm{hit}}\cdot p_{\mathrm{rx}}$
\end_inset

 should be equal to the bulk reaction rate 
\begin_inset Formula $\kappa\rho_{1}\rho_{2}\Delta t$
\end_inset

, we have 
\begin_inset Formula \[
p_{\mathrm{rx}}=\frac{4\,\kappa}{3\, D\, A\, B}\]

\end_inset


\end_layout

\begin_layout Standard
if only species 2 diffuses (note that 
\begin_inset Formula $\lambda^{2}/\Delta t=4D$
\end_inset

).
 If both volume molecules diffuse, the total number of hits in a time 
\begin_inset Formula $\Delta t$
\end_inset

 is 
\begin_inset Formula \[
\frac{\Delta t}{\Delta t_{1}}n_{\mathrm{hit},1}+\frac{\Delta t}{\Delta t_{2}}n_{\mathrm{hit},2}=\frac{3}{16}\left(4D_{1}+4D_{2}\right)\rho_{1}\rho_{2}AB\Delta t\]

\end_inset

 so that 
\begin_inset Formula \[
p_{\mathrm{rx}}=\frac{4\,\kappa}{3\, A\, B\,(D_{1}+D_{2})}\]

\end_inset

 If the surface can be hit from either side, the number of hits doubles,
 so the reaction rate should be halved: 
\begin_inset Formula \[
p_{\mathrm{rx}}^{\prime}=\frac{2\,\kappa}{3\, A\, B\,(D_{1}+D_{2})}\]

\end_inset


\end_layout

\begin_layout Subsubsection*
Three-way reactions with two surface components
\end_layout

\begin_layout Standard
The standard computation for the number of hits against a single surface
 is 
\begin_inset Formula \[
n_{\mathrm{hit}}=\frac{\rho_{1}A\lambda}{2\sqrt{\pi}}\]

\end_inset

 and the probability that a grid element is filled with an appropriate surface
 molecule is 
\begin_inset Formula $\sigma_{i}A$
\end_inset

 so that if we demand that we strike the first surface molecule directly
 and the second is adjacent, then the productive hit rate is (keeping in
 mind that either of the two surface molecules can be the initial target)
 
\begin_inset Formula \[
n_{\mathrm{productive}}=\frac{3\rho_{1}\sigma_{2}\sigma_{3}A^{3}\lambda}{\sqrt{\pi}}\]

\end_inset

 The desired number of reactions is 
\begin_inset Formula $\kappa\rho_{1}\sigma_{2}\sigma_{3}A\Delta t$
\end_inset

 (if appropriate units are used for the rate constant), so that 
\begin_inset Formula \[
p_{\mathrm{rx}}=\kappa\frac{\sqrt{\pi}}{3A^{2}v}\]

\end_inset

In contrast, if one of the components is the surface itself (let's assign
 it to 
\begin_inset Formula $\mathrm{\sigma_{3}}$
\end_inset

) and the other is the molecule, the number of productive collisions drops
 by a factor of six (no neighbors, and only one target), 
\begin_inset Formula $\sigma_{3}A=1$
\end_inset

 in the probability calculation and 
\begin_inset Formula $\sigma_{3}$
\end_inset

 is typically omitted entirely from the bulk equation, giving 
\begin_inset Formula \[
p_{\mathrm{rx}}^{\prime}=\kappa\frac{2\sqrt{\pi}}{Av}\]

\end_inset


\end_layout

\begin_layout Subsubsection*
Three-way reactions with all surface components
\end_layout

\begin_layout Standard
If all components are in the surface, the total number of reactions per
 timestep for a single molecule should be 
\begin_inset Formula $\kappa\sigma_{2}\sigma_{3}\Delta T$
\end_inset

 and the actual probability of finding the appropriate neighbors is 
\begin_inset Formula $3\sigma_{2}A\cdot2\sigma_{3}A$
\end_inset

.
 Thus the reaction probability should be 
\begin_inset Formula \[
p_{\mathrm{rx}}=\frac{\kappa}{6A^{2}}\Delta t\]

\end_inset


\end_layout

\begin_layout Subsubsection*
N-way reactions in MCell
\end_layout

\begin_layout Standard
Generalizing to 
\begin_inset Formula $N+1$
\end_inset

 reactants (one moving and 
\begin_inset Formula $N$
\end_inset

 targets), where 
\begin_inset Formula $N$
\end_inset

 is a positive integer, we find that the collision rate is 
\begin_inset Formula \[
n_{\mathrm{{hits}}}=\int_{0}^{\infty}dR\cdot\prod_{i=1}^{N}\rho_{i}\cdot\left(R\, A\right)^{N}\cdot\frac{4\pi R^{2}}{\pi^{3/2}\lambda^{3}}e^{-R^{2}/\lambda^{2}}=\prod_{i=1}^{N}\rho_{i}\cdot\frac{2\lambda^{N}A^{N}}{\sqrt{\pi}}\Gamma\left(\frac{N+3}{2}\right)\]

\end_inset

 and the bulk rate is 
\begin_inset Formula $\kappa\cdot\prod_{i=1}^{N}\rho_{i}\cdot\Delta t$
\end_inset

, so equating probabilities gives 
\begin_inset Formula \[
p=\frac{\kappa\sqrt{\pi}\Delta t}{2\lambda^{N}A^{N}\Gamma\left(\frac{N+3}{2}\right)}\]

\end_inset

 Note that 
\begin_inset Formula $\Gamma(N)=(N-1)!$
\end_inset

 and 
\begin_inset Formula $\Gamma(N+\frac{1}{2})=\sqrt{\pi}\cdot2^{-2N}\cdot(2N)!/N!=\sqrt{\pi}\cdot\prod_{i=1}^{N}\frac{2i-1}{2}$
\end_inset

.
 If we have multiple moving molecules, 
\begin_inset Formula \[
\kappa\cdot\prod_{i=1}^{N+1}\rho_{i}\cdot\Delta t=\prod_{i=1}^{N+1}\rho_{i}\cdot\frac{2A^{N}}{\sqrt{\pi}}\Gamma\left(\frac{N+3}{2}\right)\cdot\sum_{i=1}^{N+1}p_{i}\frac{\Delta t}{\Delta t_{i}}\lambda_{i}^{N}\]

\end_inset

 so that, if we set all the 
\begin_inset Formula $p_{i}$
\end_inset

to be equal, 
\begin_inset Formula \[
p=\frac{\kappa\sqrt{\pi}}{2\Gamma\left(\frac{N+3}{2}\right)A^{N}\sum\frac{\lambda_{i}^{N}}{\Delta t_{i}}}\]

\end_inset

 By induction on the result for pairs of targets in the 3-way case, we also
 see that if a target 
\begin_inset Formula $i$
\end_inset

 is hit in a restricted space, the reaction probability should be multiplied
 by 
\begin_inset Formula $A/A_{i}^{\star}$
\end_inset

.
\end_layout

\begin_layout Subsubsection*
Higher order reactions with single surfaces
\end_layout

\begin_layout Standard
The primary equation for 
\begin_inset Formula $p_{\mathrm{hit}}(h)$
\end_inset

 remains the same for higher-order reactions except that 
\begin_inset Formula $\rho_{1}Ar$
\end_inset

 turns into 
\begin_inset Formula $\prod_{i=1}^{N-1}\rho_{i}Ar$
\end_inset

 (recall that 
\begin_inset Formula $r=\sqrt{r_{xy}^{2}+r_{z}^{2}}$
\end_inset

.
 Otherwise the integration is the same.
 The general formulae for such integrals is rather tricky, but the first
 few values for 
\begin_inset Formula $n_{\mathrm{hit}}$
\end_inset

 are listed here: 
\begin_inset Formula \[
\begin{array}{rcl}
n_{\mathrm{hit}}^{(N=2)} & = & \frac{3}{16}\lambda^{2}AB\prod_{i=1}^{N}\rho_{i}\\
n_{\mathrm{hit}}^{(N=3)} & = & \frac{1}{2\sqrt{\pi}}\lambda^{3}A^{2}B\prod_{i=1}^{N}\rho_{i}\\
n_{\mathrm{hit}}^{(N=4)} & = & \frac{15}{32}\lambda^{4}A^{3}B\prod_{i=1}^{N}\rho_{i}\\
n_{\mathrm{hit}}^{(N=5)} & = & \frac{3}{2\sqrt{\pi}}\lambda^{5}A^{4}B\prod_{i=1}^{N}\rho_{i}\\
n_{\mathrm{hit}}^{(N=6)} & = & \frac{105}{64}\lambda^{5}A^{5}B\prod_{i=1}^{N}\rho_{i}\\
n_{\mathrm{hit}}^{(N=7)} & = & \frac{6}{\sqrt{\pi}}\lambda^{6}A^{5}B\prod_{i=1}^{N}\rho_{i}\end{array}\]

\end_inset

 The author conjectures that the formula for even 
\begin_inset Formula $N$
\end_inset

 is 
\begin_inset Formula \[
\frac{(N+1)!}{4\cdot2^{3N/2}(N/2)!}\lambda^{N}A^{N-1}B\prod_{i=1}^{N}\rho_{i}\]

\end_inset

 and for odd 
\begin_inset Formula $N$
\end_inset

 is 
\begin_inset Formula \[
\frac{((N+1)/2)!}{4\sqrt{\pi}}\lambda^{N}A^{N-1}B\prod_{i=1}^{N}\rho_{i}\]

\end_inset

 These have been checked up to 
\begin_inset Formula $N=12$
\end_inset

 in Maple 10, but these formulae have not been proven.
\end_layout

\begin_layout Subsubsection*
Higher order reactions with multiple surface components
\end_layout

\begin_layout Standard
Each time one adds a molecular surface component, one adds a factor of 
\begin_inset Formula $\frac{1}{A}$
\end_inset

 the first time (if one is already hitting a surface and a molecule needs
 to be there also), 
\begin_inset Formula $\frac{1}{3A}$
\end_inset

 the second time, 
\begin_inset Formula $\frac{1}{2A}$
\end_inset

 the third time, and 
\begin_inset Formula $\frac{1}{A}$
\end_inset

 the fourth time.
 More than four molecules cannot be found by adjacent search; if this is
 generalized to a wider search, then if the extra partner can be found in
 one of 
\begin_inset Formula $n$
\end_inset

 places, the reaction probability changes by 
\begin_inset Formula $\frac{1}{nA}$
\end_inset

.
 In addition, if there are 
\begin_inset Formula $k$
\end_inset

 possible surface targets total, the reaction probability should be multiplied
 by an additional 
\begin_inset Formula $\frac{1}{k}$
\end_inset

.
\end_layout

\begin_layout Subsubsection*
Converting 3-way reactions to 2-way reactions
\end_layout

\begin_layout Standard
Suppose we have a three-way reaction 
\begin_inset Formula \[
A_{1}+A_{2}+A_{3}\overset{k}{\longrightarrow}A_{123}\]

\end_inset

 which we wish to approximate by nine bimolecular reactions: 
\begin_inset Formula \[
\begin{array}{rcl}
A_{1}+A_{2} & \overset{k_{12}}{\longrightarrow} & A_{12}\\
A_{1}+A_{3} & \overset{k_{13}}{\longrightarrow} & A_{13}\\
A_{2}+A_{3} & \overset{k_{23}}{\longrightarrow} & A_{23}\\
A_{12} & \overset{k_{-12}}{\longrightarrow} & A_{1}+A_{2}\\
A_{13} & \overset{k_{-13}}{\longrightarrow} & A_{1}+A_{3}\\
A_{23} & \overset{k_{-23}}{\longrightarrow} & A_{2}+A_{3}\\
A_{3}+A_{12} & \overset{k_{3}}{\longrightarrow} & A_{123}\\
A_{2}+A_{13} & \overset{k_{2}}{\longrightarrow} & A_{123}\\
A_{1}+A_{23} & \overset{k_{1}}{\longrightarrow} & A_{123}\end{array}\]

\end_inset

 At quasi-steady state, we want to match rate of entry in to the 
\begin_inset Formula $A_{123}$
\end_inset

 state: 
\begin_inset Formula \begin{equation}
kA_{1}A_{2}A_{3}=k_{1}A_{1}A_{23}+k_{2}A_{2}A_{13}+k_{3}A_{3}A_{12}\label{eqn_qss_match}\end{equation}

\end_inset

 And we also wish to keep the quasi-steady state concentrations of the intermedi
ates 
\begin_inset Formula $A_{12}$
\end_inset

, 
\begin_inset Formula $A_{13}$
\end_inset

, and 
\begin_inset Formula $A_{23}$
\end_inset

 low compared to the starting materials.
 In general, we will have 
\begin_inset Formula \[
\frac{d}{dt}A_{hi}=-k_{j}A_{j}A_{hi}-k_{-hi}A_{hi}+k_{hi}A_{h}A_{i}\approx0\]

\end_inset

 so that 
\begin_inset Formula \[
A_{hi}\approx\frac{k_{hi}A_{h}A_{i}}{k_{-hi}+k_{j}A_{j}}\]

\end_inset

 If we want this to be roughly independent of the concentration of 
\begin_inset Formula $A_{j}$
\end_inset

 then we require 
\begin_inset Formula $k_{-hi}\gg k_{j}A_{j}$
\end_inset

 and can rewrite this as 
\begin_inset Formula \[
A_{hi}\approx\frac{k_{hi}}{k_{-hi}}A_{h}A_{i}\left(1-\frac{k_{j}A_{j}}{k_{-hi}}\right)\]

\end_inset

If we further require that 
\begin_inset Formula $A_{hi}$
\end_inset

 be small compared to 
\begin_inset Formula $A_{h}$
\end_inset

 and 
\begin_inset Formula $A_{i}$
\end_inset

, we also require 
\begin_inset Formula $\frac{k_{hi}}{k_{-hi}}\ll\frac{1}{\max(A_{h},A_{i})}$
\end_inset

.
 Let 
\begin_inset Formula $A_{+}$
\end_inset

 be the largest value of any of the 
\begin_inset Formula $A_{i}$
\end_inset

 during a simulation.
 Furthermore, let us set all 
\begin_inset Formula $k_{j}$
\end_inset

 to be 
\begin_inset Formula $k^{\star}$
\end_inset

, all 
\begin_inset Formula $k_{hi}$
\end_inset

 to be 
\begin_inset Formula $k^{\dagger}$
\end_inset

 and all 
\begin_inset Formula $k_{-hi}$
\end_inset

 to be 
\begin_inset Formula $k^{\ddagger}$
\end_inset

.
 Then our constraints require that 
\begin_inset Formula $k^{\ddagger}\gg k^{\star}A_{+}$
\end_inset

 and 
\begin_inset Formula $k^{\dagger}\ll k^{\ddagger}\frac{1}{A_{+}}$
\end_inset

; taken together, 
\begin_inset Formula $k^{\dagger}\approx k^{\star}$
\end_inset

 is a valid solution, so we may as well make the two the same, 
\begin_inset Formula $k^{\prime}$
\end_inset

.
 Thus, we have a forward reaction rate 
\begin_inset Formula $k^{\prime}$
\end_inset

 for all binding reactions and a backward reaction rate 
\begin_inset Formula $k^{\ddagger}$
\end_inset

 for dissociation of the intermediates.
\end_layout

\begin_layout Standard
Thus, equation (
\begin_inset LatexCommand \ref{eqn_qss_match}

\end_inset

) becomes 
\begin_inset Formula \[
kA_{1}A_{2}A_{3}\approx k^{\prime}A_{1}\frac{k^{\prime}}{k^{\ddagger}}A_{2}A_{3}+k^{\prime}A_{2}\frac{k^{\prime}}{k^{\ddagger}}A_{1}A_{3}+k^{\prime}A_{3}\frac{k^{\prime}}{k^{\ddagger}}A_{1}A_{2}=3\frac{{k^{\prime}}^{2}}{k^{\ddagger}}A_{1}A_{2}A_{3}\]

\end_inset

 with a first-order error term 
\begin_inset Formula \[
-\frac{{k^{\prime}}^{3}}{{k^{\ddagger}}^{2}}A_{1}A_{2}A_{3}\left(A_{1}+A_{2}+A_{3}\right)\]

\end_inset

 If we let 
\begin_inset Formula $k^{\ddagger}=\alpha k^{\prime}$
\end_inset

, where 
\begin_inset Formula $\alpha\gg A_{+}$
\end_inset

, we then have 
\begin_inset Formula \[
k\approx\frac{3}{\alpha}k^{\prime}-\frac{A_{1}+A_{2}+A_{3}}{\alpha^{2}}k^{\prime}\]

\end_inset

 Thus, 
\begin_inset Formula \[
k_{1}=k_{2}=k_{3}=k_{12}=k_{13}=k_{23}=k^{\prime}\approx\frac{1}{3}\alpha k\]

\end_inset

 and 
\begin_inset Formula \[
k_{-12}=k_{-13}=k_{-23}=k^{\ddagger}\approx\frac{1}{3}\alpha^{2}k\]

\end_inset

 Note that our fractional error is approximately 
\begin_inset Formula $1/\alpha$
\end_inset

, i.e.
 if we let 
\begin_inset Formula $\alpha=100\, A_{+}$
\end_inset

 our fractional error would be under 1%.
\end_layout

\begin_layout Subsubsection*
Notes on Units
\end_layout

\begin_layout Standard
When rates are measured for bimolecular reactions between a volume molecule
 and a surface molecule, one can lay down a surface with known (or measurable)
 area 
\begin_inset Formula $\mathcal{A}$
\end_inset

 in a solution of volume 
\begin_inset Formula $\mathcal{V}$
\end_inset

.
 You then add 
\begin_inset Formula $n_{1}$
\end_inset

 volume molecules (concentration 
\begin_inset Formula $\rho_{1}=n_{1}/\mathcal{V}$
\end_inset

 in units of #/unit volume) and 
\begin_inset Formula $n_{2}$
\end_inset

 surface molecules (at density 
\begin_inset Formula $\sigma_{2}=n_{2}/\mathcal{A}$
\end_inset

) and measure 
\begin_inset Formula \[
\frac{d\rho_{1}}{dt}=-k_{\rho}\rho_{1}\sigma_{2}\]

\end_inset


\end_layout

\begin_layout Standard
where 
\begin_inset Formula $k_{\rho}$
\end_inset

 is the rate constant with units of 
\begin_inset Formula $\mathrm{area}\cdot\#^{-1}\cdot\mathrm{s}^{-1}$
\end_inset

.
 One can equally well write this as 
\begin_inset Formula \[
\frac{d\sigma_{2}}{dt}=-k_{\sigma}\rho_{1}\sigma_{2}\]

\end_inset

 where 
\begin_inset Formula $k_{\sigma}$
\end_inset

 has units of 
\begin_inset Formula $\mathrm{volume}\cdot\#^{-1}\cdot\mathrm{s}^{-1}$
\end_inset

.
 Of course, the numbers of molecules reacting are the same, so that 
\begin_inset Formula \[
-k_{\sigma}\mathcal{V}^{-1}n_{1}n_{2}=\mathcal{A}\frac{d\sigma_{2}}{dt}=\frac{dn_{2}}{dt}=\frac{dn_{1}}{dt}=\mathcal{V}\frac{d\rho_{1}}{dt}=-k_{\rho}\mathcal{A}^{-1}n_{1}n_{2}\]

\end_inset

 We can now let 
\begin_inset Formula $k_{n}=k_{\sigma}\mathcal{V}^{-1}=k_{\rho}\mathcal{A}^{-1}$
\end_inset

 and write 
\begin_inset Formula \[
\frac{dn_{\star}}{dt}=-k_{n}n_{1}n_{2}\]

\end_inset


\end_layout

\begin_layout Standard
But one can also define 
\begin_inset Formula $\rho_{2}=\sigma_{2}\cdot\frac{\mathcal{A}}{\mathcal{V}}$
\end_inset

, that is, treat the surface molecule as if it were a volume molecule, and
 then 
\begin_inset Formula \[
\frac{d\rho_{\star}}{dt}=\frac{dn_{\diamond}}{dt}\mathcal{V}^{-1}=-k_{n}\rho_{1}\rho_{2}\mathcal{V}^{-1}=-k_{\star}\rho_{1}\rho_{2}\]

\end_inset

 where 
\begin_inset Formula $k_{\star}=k_{n}\cdot\mathcal{V}^{-1}$
\end_inset

; here 
\begin_inset Formula $\star$
\end_inset

 stands for one of 
\begin_inset Formula $1$
\end_inset

 or 
\begin_inset Formula $2$
\end_inset

, while 
\begin_inset Formula $\diamond$
\end_inset

 stands for the other.
\end_layout

\begin_layout Standard
If one is performing a stochastic calculation, the total number of hits
 on all surface molecules in a short time 
\begin_inset Formula $\Delta t$
\end_inset

 is 
\begin_inset Formula \[
n_{2}\frac{\rho_{1}A\lambda}{2\sqrt{\pi}}\Delta t\]

\end_inset

 where 
\begin_inset Formula $A$
\end_inset

 is the area of a single surface molecule.
 From the well-mixed continuum approximation, the probability should be
 scaled such that 
\begin_inset Formula \[
-p_{\mathrm{rx}}\cdot n_{2}\frac{\rho_{1}A\lambda}{2\sqrt{\pi}}=\frac{dn_{\diamond}}{dt}\Delta t=\mathcal{V}\frac{d\rho_{\star}}{dt}\Delta t=-k_{\star}n_{2}\rho_{1}\Delta t\]

\end_inset

 so that 
\begin_inset Formula $p_{\mathrm{rx}}=k_{\star}\cdot2\sqrt{\pi}/Av$
\end_inset

 where 
\begin_inset Formula $v=\lambda/\Delta t$
\end_inset

.
 Thus, we can use the volumetric rate constant 
\begin_inset Formula $k_{\star}$
\end_inset

 where we need only convert from molarity to #/unit volume---we need not
 know the original volume of the test sample or the area of membrane in
 it, as long as the value of 
\begin_inset Formula $k_{\mathrm{\star}}$
\end_inset

 is reported.
 Conveniently, one can measure 
\begin_inset Formula $k_{\star}$
\end_inset

 without even knowing the area of the membrane.
\end_layout

\begin_layout Standard
However, if we add a second surface component at density 
\begin_inset Formula $\sigma_{3}$
\end_inset

, the above is no longer true since the reaction rate is no longer proportional
 to the numbers of each molecule.
 In particular, 
\begin_inset Formula \[
\frac{d\rho_{1}}{dt}=-k_{\rho}\rho_{1}\sigma_{2}\sigma_{3}\]

\end_inset

 
\begin_inset Formula \[
\frac{d\sigma_{i\in\{2,3\}}}{dt}=-k_{\sigma}\rho_{1}\sigma_{2}\sigma_{3}\]

\end_inset

 defines the reactions, but now 
\begin_inset Formula \[
\frac{dn_{1}}{dt}=\mathcal{V}\frac{d\rho_{1}}{dt}=-k_{\rho}n_{1}n_{2}n_{3}\mathcal{A}^{-2}\]

\end_inset

 
\begin_inset Formula \[
\frac{dn_{i}}{dt}=\mathcal{A}\frac{d\sigma_{i}}{dt}=-k_{\sigma}n_{1}n_{2}n_{3}\mathcal{A}^{-1}\mathcal{V}^{-1}\]

\end_inset

 so that 
\begin_inset Formula $k_{n}=k_{\sigma}\mathcal{A}^{-1}\mathcal{V}^{-1}=k_{\rho}\mathcal{A}^{-2}$
\end_inset

.
 If we try the same trick of converting 
\begin_inset Formula $\sigma_{2}$
\end_inset

 and 
\begin_inset Formula $\sigma_{3}$
\end_inset

 to volumes, we find that -k
\begin_inset Formula \[
-k_{\star}\rho_{1}\rho_{2}\rho_{3}=\frac{d\rho_{\star}}{dt}=\mathcal{V}^{-1}\frac{dn_{\star}}{dt}=-k_{n}n_{1}n_{2}n_{3}\mathcal{V}^{-1}=-k_{n}\rho_{1}\rho_{2}\rho_{3}\mathcal{V}^{2}\]

\end_inset

 so that 
\begin_inset Formula $k_{n}=k_{\star}\mathcal{V}^{-2}$
\end_inset

.
 When we try to match total numbers of molecules reacting, we find that
 there are 
\begin_inset Formula \[
p_{\mathrm{rx}}\frac{3\rho_{1}\sigma_{2}\sigma_{3}A^{3}\lambda}{\sqrt{\pi}}\cdot\frac{\mathcal{A}}{A}\]

\end_inset

 total reactions in time 
\begin_inset Formula $\Delta t$
\end_inset

 in a stochastic treatment (the factor of 
\begin_inset Formula $\mathcal{A}/A$
\end_inset

 arises from the difference between the per-receptor area and the total
 surface area), and 
\begin_inset Formula \[
\frac{dn_{\star}}{dt}\Delta t=\mathcal{V}\frac{d\rho_{\star}}{dt}\Delta t=-k_{\star}\rho_{1}\rho_{2}\rho_{3}\mathcal{V}\Delta t=-k_{\star}\rho_{1}\sigma_{2}\sigma_{3}\frac{\mathcal{A}^{2}}{\mathcal{V}}\Delta t\]

\end_inset

 from the deterministic continuum equations.
 Equating the two (with the correct sign) gives 
\begin_inset Formula \[
p_{rx}=k_{\star}\frac{\sqrt{\pi}}{3A^{2}v}\cdot\frac{\mathcal{A}}{\mathcal{V}}\]

\end_inset

 This is problematic because the probability of reaction now depends on
 the surface to volume ratio 
\begin_inset Formula $\mathcal{A}/\mathcal{V}$
\end_inset

; what we need is 
\begin_inset Formula $k_{\star}\frac{\mathcal{A}}{\mathcal{V}}$
\end_inset

.
 Fortunately, 
\begin_inset Formula $k_{\sigma}=k_{n}\mathcal{A}\mathcal{V}=k_{\star}\frac{\mathcal{A}}{\mathcal{V}}$
\end_inset

.
 Thus, the only appropriate rate constant for three-molecule reactions is
 
\begin_inset Formula $k_{\sigma}$
\end_inset

, which has units of 
\begin_inset Formula $\mathrm{volume}\cdot\mathrm{area}\cdot\#^{-2}\cdot s^{-1}$
\end_inset

.
\end_layout

\end_body
\end_document