Conversion model

We developed the conversion model to predict the amount of the cellulose needed to be hydrolyzed, and thus predicted the concentration of cellobiose we got during the experiment . The model was to optimize the substrate cellulose input as the material to manufacture bacterial cellulose. Since an excessive amount of cellobiose could drastically decrease the activity of endoglucansse(CenA) and exoglucansse(Cex), we must chose a appropriate amount of wastepaper pulp containing suitable cellulose content.The prediction was necessary since different cellulose concentrations would result in different concentrations of cellobiose(Zhuoliang Ye.2011). We assumed parallel concentrations to investigate the conversion rate of cellulose and by comparing the data collected from the references(Zhuoliang Ye,R.2014), we modified several relevant parameters with Langmuir adsorption equation and modeled Michaelis-Menten equation(Yi Zheng.2016).

By doing so, we expected to get the highest concentration of cellobiose with the expression of endoglucanase and exoglucanase and the suitable time of fermentation to open the switch of next stage and offered a guidance to our wet experiment.We found the most suitable concentration of cellulose was 20g/L and time of fermentation was 24hours and we got 6g/L cellobiose(Yi Zheng.2016).

Here we listed some equations used for our further modeling

k_f:inactivation rate constant for adsorbed enzyme;
k_r:reactivation rate constant;
A_max:maximum adsorption sites per unit substrate;
K_d:equilibrium constant of dissociation;
K_m:Michaelis constant;
V_r:actual reaction rate;
A₀:k_f /(k_f+k_r);
y₀:k_r /(k_f+k_r);
t_1/2:ln2/(k_f+k_r);

During experiments, we wanted to estimate the final concentration of our intermediate cellobiose to make sure this concentration reaches the value for activating cellobiose operon(Maxime Toussaint.2016)(Aashiq H.2007), but we have reached the stage of fermentation, so modeling served as a most convenient tool to predict for our purpose(Plumbridge, J.2004).

A₁、A₂、t₁、t₂: empirical parameter;

Based on the above results, we also combined our measurement results of the enzyme activity of CenA and Cex(Q Gan.2003), and the range of final concentration against ranging enzyme activity was predicted(Prabuddha Bansal.2009).

f: fractal dimension
k₂: product formation rate constant
[E]: enzyme concentration
[P]: product concentration
[S]: substrate concentration
K_m: Michaelis constant

Enzymatic activity adjustment model

We developed an enzymatic activity adjustment model to predict the most desired pH and temperature condition to balance the inactivation of cellulase with maintaining stable proceeding of cellulose synthesis(Kwabena O.2016). Since the cellulase and cellulose synthase were functionally antagonistic with each other, denoting the latter produced bacterial cellulose might be easily degraded if the former secreted cellulase were not effectively inhibited. Nevertheless, adding cellulose inhibitor and regulating the reaction condition were both feasible to decrease cellulase activity, the latter one seemed to be a more economic method(Damude H G.1996).

We performed numerical simulations on the model and sensitivity analysis on some key parameters, and design several reaction conditions to test the accuracy of the equations(Honda Yuji.2005). By conducting wet experiment on CMCNase activity test, we got the specific activity data of cellulase and compared them with the simulated one to adjust some key factors to better conform to our engineered bacteria(Nikolova P V.1997).

Finally, we estimated the best condition to meet this double demands was at pH6.5 and temperature 28℃, in this case, the specific activity of cellulase could drop to 15% ~ 10% of the normal activity at pH7 and temperature 37℃.

Fermentation model

To ascertain the suitable strain concentration and enzyme activity,we.built follow models.Genes of endocellulase and exocellulase expressed conducted by λ promoter at first.Without inhibition for λ promoter,genes of endocellulase and exocellulase transcribed and translated as E.coli strain grew.So we aimed to incubate E.coli at liquid medium and predicted the concentration of bacteria and activity of enzymes with Logistic equation and Luedeking-Piret equation(Garnier Alain.2015). As a result,we assumed the relationship between concentration of E.coli and enzymes was semi-related and predicted how much we could get enzymes.After incubating 24 hours,the E.coli concentration(OD) would rise to 35 and enzyme activity would ascend to 50 U/mL in the fermentor.

X: E.coli concentration(OD);
μ_max:maximum specific growth rate(h-1);
X_m:maximum strain concentration(OD);
t:time(h);

C_p: cellulase activity(U/mL);
C_x:strain concentration(g/L);
a:growth-relative coefficient(U/mg);
t:time(h);

Product model

We had this model to clear how much sugar was used to produce bacteria cellulose.When cellobiose in fermentation liquor entered E.coli, with expression of Cellobiose Phosphorylase and Bacterial Cellulose Synthase, the strain began to produce bacterial cellulose utilizing cellobiose. So, we analyzed the metabolic pathways from glucose to UDPG and from UDPG to bacteria cellulose(Tikhonova.2018), predicting the varieties of the concentration of cellobiose and bacteria cellulose.The metabolic pathway of UDPG would transfer about 5% of sugar including cellobiose and other sugars to BC and we could get 1.3g/L cellobiose in the end.