Derivation

Using parameters shown in Table 1, and simplifying the S.cerevisiae cell as a sphere, and E.coli cell as a cylinder, we calculate the solid angle (Ω) of both of them.

\[\Omega=\iint_S{\sin\theta d\theta d\phi}\]

Therefore, the solid angle of a S.cerevisiae and an SEi could be given.

\[\Omega_S=4\pi\] \[\Omega_{SE_i}\approx0.11\]

The proportion of occupied solid angle could be calculated.

\[\frac{\Omega_{SE_i}}{\Omega_S}=0.96\%\lt1\%\]

Although the primary binding process may affect the secondary binding process, the proportion of occupied solid angle is less than 1%, which means assuming binding and dissociation process as independent events is reasonable. Independent events keep the same probability all the time, so that we could calculate the probabilities of both of them.

Binding Probability(Probb)

Given that binding only occurs when a SE_i meets with an E.coli cell, the sample space of the binding probability is considered as “A SE_i and an E.coli cell meeting each other”.

\[Probb=P("B"\mid"EM") P("EM")\]

where “B” equals “Binding” and “EM” equals “Effective meeting”. “Effective meeting”, in other word, is the streptavidin and biotin meeting each other as two cells encounter. Therefore, alternative form if Probb is given.

\[Probb=\alpha \frac{N_{bio} A_{bio}}{A_{S.cere}}\cdot\frac{N_{strep} A_{strep}}{A_{E.co}}\]

By bringing parameters in Table 1, Probb is 54.503 % in our model.

Dissociation Probability(Probd)

Not like binding process, which may happens only when two cells encounter, the dissociation process could be taken place at any time, so the Probd is defined as the probability of dissociation in an unit time interval (Δt) following the exponential distribution.

\[Probd=1-{exp} (-k_d \Delta t)\]

By bringing parameters in Table 1, Probd is 0.339 % in our model.

*Supplementary: Derivation of Probd function
The dissociation process could be simplified as follows:

\[SE_i\to SE_{i-1}+E \]

And we could write down the ODE function of it.

\[\frac{-d[SE_i]}{dt}=k_d[SE_i]\]

By integrating it with \(t:0\to\Delta t\), \([SE_i]:[SE_i]_0\to[SE_i]_0+\Delta[SE_i]\), we have

\[\ln\left(\frac{[SE_i]_0-\Delta[SE_i]}{[SE_i]_0}\right)=-k_d\Delta t\]

Alternative form of this equation is as follows.

\[\frac{\Delta[SE_i]}{[SE_i]_0}=1-exp(-k_d\Delta t)\]

From the definition of Probd, we could find that

\[Probd=\frac{\Delta[SE_i]}{[SE_i]_0}\]

And we give the probability of dissociation.

\[Probd=1-exp(-k_d\Delta t)\]