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.

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

Derivation of ANRC

By adopting the Monod function and stoichiometry rules in our adhesion platform, the production could be simplified as:

The production rate of ethanol is given.

\[V_{eth}=\frac{d[ethanol]}{dt}=\sum^n_{i=0}SE_i\left(\frac{k_i[xylose]}{k_i+[xylose]}\right)\]

The alternative form of this function is:

\[V_{eth}=S\times\sum^n_{i=0}\alpha_i\left(\frac{k_i[xylose]}{k_i+[xylose]}\right)\]

If we consider \(v_{eth}\) as the addition of i \(v_{ethi}\), we would get

\[V_{eth_i}=S\alpha_i\times\left(\frac{k_i[xylose]}{k_i+[xylose]}\right)\]

We could find that the production rate constant of \(v_{ethi}\) is the latter item in function, which we defined as \(r_i\).

\[r_i=\left(\frac{k_i[xylose]}{k_i+[xylose]}\right)\]

This means the factors determining the rate constant of each type of SEi are (1) \(k_i\),\(K_i\) the characteristics of SE_i which are the same for all type of SE_i, because their ethanol production unit are S.cerevisiae cell. (2) [xylose] the partial concentration of the xylose surrounding each cell, and this is the difference between different types of SE_i, because the partial concentrations of SE_i is higher than that of SE_i 1.

The partial concentration of xylose for SE_i could be divided into two parts —— a part produced from E.coli cells binding it ([xylose]_self), and a part from other E.coli cells as background ([xylose]_bg).

\[[xyloe]=[xylose]_{self}+[xyloe]_{bg}\]

[xylose]_bg is the stable concentration of xylose in this system, and is determined by its production rate and consumption rate. Easy to notice that if [xylose]_bg is high enough, the concentration of xylose ([xylose]) would be nothing different between adhesion platform and non-adhesion platform, which means the advantage of adhesion platform would not be so significant. Therefore, to overcome it, one possible solution is to accelerate the process of consuming xylose by increasing the transportation rate of the transporters on S.cerevisiae cells to decrease the [xylose]_bg. It seems that the transporter is the key factor of making adhesion platform significant.

[xylose]_self is the characteristic property of each type of SE_i that could not be affected by environments. Easy to find that simulations in “Production Properties of SE_i” are to get r_i values by setting the amount of S.cerevisiae cells to be 1.

By normalizing the r_i values to the maximum of them, we have the normalized production rate constant p_i.

\[p_i=\frac{r_i}{max(r_i)}\]

So the normalized ethanol production rate is as follows:

\[[Normalized_{V_{eth}}]=S\times\sum^n_{i=0}(a_i\times p_i)\]

To eliminate the distinctions resulting from different amounts of cells in different systems, we take the average of the [Normalized v_eth] to obtain the mean level of ethanol production rate per cell, and defined it as Average Normalized Rate Constant (ANRC).

And the equation of ANRC is also given:

\[ANRC=\frac{S}{S+E}\sum^n_{i=0}(p_i\times a_i)\]

Wondering whether our adhesion platform would have a better performance than non-adhesion platform, we defined Δ.

\[\Delta=ANRC_{adhe}-ANRC_{non-adhe}\]

From the definition of ANRC, easy to get \(ANRC_{non-adhe}\) .

\[ANRC_{non-adhe}=p_0\times\frac{S}{S+E}\]

Then the function of Δ is as follows:

\[\Delta=\frac{S}{S+E}\sum^n_{i=0}[(p_i-p_0)\times a_i]\]