Hill function of activator

$$f(X) = \frac{\beta X ^ n}{K ^ n + X ^ n}$$

Hill function, illustrating the process which the ligand (X) bound to its corresponding receptor, has three key parameters: K, β and n. The first parameter K, known as activation coefficient with a concentration unit. It defines the needed concentration of X in activation state which can turn on gene expression effectively. Meanwhile, the specified value of K is depended on chemical affinity between X and its receptor. The second parameter β says the biggest expression level of promoter. Finally, Hill coefficient n determines the degree of steep input function (slope).

Michaelis-Menten equation

$$E + S\underset{{k_{-1}}}{\stackrel{k_{1}} \rightleftharpoons} ES \to P + E$$ $$V=\frac{V m[S]}{K m+[S]}$$

Michaelis-Menten equation in the case of noncompetitive inhibition

\begin{array}{l} \;E + S\underset{{k_{-1}}}{\stackrel{k_{1}} \rightleftharpoons} ES \to P + E \\ \;+ \\ \;I\\ \,\downarrow \small k_{i} \\ EI \end{array} $$V=\frac{V m[S]}{(K m+[S])(1+\frac{[I]}{K_{i}} )} $$

Michaelis-Menten equation, the rate equation for a one substrate enzyme-catalyzed reaction, describes the hyperbolic dependence of the initial reaction velocity, V₀, on substrate concentration [S] and inhibitor concentration [I]. Reaction speed depends on two parameters. V_m defines the maximal velocity of a reaction and occurs at high substrate concentrations when the enzyme is saturated. K_m is known as Michaelis constant which is the concentration of substrate when reaction speed reach half of maximum velocity.

Logistic equation

The logistic function was first introduced as a model describing population growth, and gradually applied in a range of fields.

$$ f(X) = \frac{L}{1 + e^{-k(x-x_{0} )}} $$

In our model, we use modified logistic function to illustrate the degradation kinetics.

$$S=S_{0}\left[1-\frac{1}{1+e^{\frac{4 R_{m}}{S_{0}}(\lambda-t)+2}}\right]$$

S is substrate concentration at time t (g L^-1), t is incubation time (h), S₀ is initial substrate concentration (g L^-1), λ is lag time (h), R_m is maximum substrate degradation rate (g L^-1 h^-1) ².

In our light-on system, photosensitive transcription factor GAVPO will be dimerized to bind to the UASG sequence in the target transcription unit after blue light irradiation and then activate the expression of Cex, CenA and Bglx which can produce exo-1,4-D-glucanase and 1,4-D-glueanohydrolase and β-glucosidase to degrade the cellulose scaffold. Here, we quantitively visualize the expression of each enzyme, providing a hint for the possible degradation time.

Model Assumptions

Since inactive GAVPO state presents stability against denaturation even during chemical attack and GAVPO aggregates can exhibit significant dimer content ⁴, we make the following assumptions:
(1) Aggregation of inactive GAVPO does not occur.
(2) GAVPO lit monomer state is ignored due to its high convertibility.
(3) The formation of GAVPO dimer form GAVPO monomer is irreversible ³.

Result

The model for the aggregation of GAVPO roughly involves two main processes: GAVPO monomers are activated and aggregates under blue light irradiation. And, the activated GAVPO initiated the downstream gene expression. Meanwhile, the GAVPO monomers / dimers will be degraded.
The reactions are described as follows:

$$ \begin{array}{l} GAVPO \; Gene \stackrel{k_{0}}{\rightarrow}{ } inactive \; GAVPO \\ inactive \; GAVPO \stackrel{k_{1}}{\rightarrow} active \; GAVPO\\ 2 \quad active \; GAVPO \stackrel{k_{2}}{\rightarrow} GAVPO \; dimer \\ GAVPO \; dimer \rightarrow exo\,\mbox{-}1,4\,\mbox{-}D\,\mbox{-}glucanase \\ GAVPO \; dimer \rightarrow 1,4\,\mbox{-}D\,\mbox{-}glueanohydrolase \\ GAVPO \; dimer \rightarrow \beta\,\mbox{-} glucosidase\\ exo\,\mbox{-}1,4\,\mbox{-}D\,\mbox{-}glucanase \stackrel{k_{3}}{\rightarrow} \phi \\ 1,4\,\mbox{-}D\,\mbox{-}glueanohydrolase \stackrel{k_{6}}{\rightarrow} \phi \\ \beta\,\mbox{-} glucosidase \stackrel{k_{7}}{\rightarrow} \phi \\ active \; GAVPO \stackrel{k_{4}}{\rightarrow} \phi \\ inactive \; GAVPO \stackrel{k_{5}}{\rightarrow} \phi \end{array} $$

We modeled reactions with the following set of ODEs.

$$ \begin{array}{l} \left.\frac{d[inactive \; GAVPO]}{d t}=k_{0}-k_{1}[active \; GAVPO]-k_{5} [inactive \; GAVPO]\right.\\ \frac{d[active \; GAVPO]}{d t}=k_{1}[inactive \; GAVPO]-k_{2}[active \; GAVPO]^{2} \\ \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\;\, -k_{5}[inactive \; GAVPO] \\ \frac{d[GAVPO \; dimer]}{d t}=k_{2}[active \; GAVPO]^{2}-k_{6}[GAVPO \; dimer] \\ \frac{d[exo\,\mbox{-}1,4\,\mbox{-}D\,\mbox{-}glucanase]}{d t}=\frac{\beta \; [GAVPO \; dimer]^{n}}{K_{1}^{n_{1}}+[GAVPO \; dimer]^{n_{1}}}-k_{3}[exo\,\mbox{-}1,4\,\mbox{-}D\,\mbox{-}glucanase] \\ \frac{d[1,4\,\mbox{-}D\,\mbox{-}glueanohydrolase]}{d t}=\frac{\beta \; [GAVPO \; dimer]^{n}}{K_{1}^{n_{1}}+[GAVPO \; dimer]^{n_{1}}}-k_{6}[1,4\,\mbox{-}D\,\mbox{-}glueanohydrolase] \\ \frac{d[\beta\,\mbox{-} glucosidase]}{d t}=\frac{\beta \; [GAVPO \; dimer]^{n}}{K_{1}^{n_{1}}+[GAVPO \; dimer]^{n_{1}}}-k_{7}[\beta\,\mbox{-} glucosidase] \\ \end{array}$$

Fig. 2 Expression of 1,4-D-glucanase

Fig. 3 Expression of of 1,4-D-glueanohydrolase

Fig. 4 Expression of β-glucosidase

According to our results, the concentration of cellulase reaches saturation at a certain point of the illumination time. We can get from the simulated results that the values of time and cellulase's concentration of saturation. We record the latter one and use it as a known quantity of the degradation model.

BC, whose chemical nature is cellulose, could not be degraded in vivo, which may cause unnecessary immune reaction after it has been used. We hope that when the wound heals to an appropriate extent, BC can be biodegraded into cellobiose or glucose, since they are soluble so that they could be metabolized by human body. Cellobiose or glucose, the decision directly affects the choice we made when choosing the hydrolase. If we set cellobiose as our final degradation product, exo-1,4-D-glucanase (Cex) and 1,4-D-glueanohydrolase (CenA) are needed. If glucose is the ultimate product, exo-1,4-D-glucanase, 1,4-D-glueanohydrolase and β-glucosidase are required. Both of these schemes have its superiority. Bluntly speaking, glucose is the energy source of our bodies; however, it will inhibit the enzymatic activity of exo-1,4-D-glucanase and 1,4-D-glueanohydrolase. Thus, degraded into cellobiose is kinetically better option, while degraded into glucose could achieve the goal of fair use. Here, we established cellulose-degradation model to estimate which kind of degraded way has quicker rate of degradation and try to explore a better option for our project.

In order to obtain more accurate results, we chose two equations, Logistic Equation and Michaelis-Menten equation, to simulate the degradation process.

Model Assumption

Since cellulose is a highly polymerized substance, it's quite difficult to measure the degradation efficiency without hypothesis its degree of polymerization (DP) when using the concentration of cellobiose or glucose as criterion. Meanwhile, exo-1,4-D-glucanase will catalyze cellulose into lower polymerization degree one. Thus, we make the following assumptions to simplify this complicated process:
(1) The degree of polymerization (DP) is 6000 if there doesn't have any additional declarations
(2) The reaction catalyzed by exo-1,4-D-glucanase, 1,4-D-glueanohydrolase are only described by one equation, which illustrate the process where cellulose is breaking down into cellobiose.

Direct fitting of logistic model

Since logistic model cannot depict the inhibit process individually; therefore, we only use logistic model to illustrate the first two step, where the final degradation product is cellobiose. The reactions of degradation of cellulose described by Logistic Equation are as follows²:

$$ S=S_{0}\left[1-\frac{1}{1+e^{\frac{4 R_{m}}{S_{0}}(\lambda-t)+2}}\right] $$

Fig. 5 Degradation described by logistic model

Our model result elegantly fits into experiment result, indicating that logistic model could beautifully represent degradation process. Surprisingly, our wet lab also observed the coordinating lagging-like phenomena, where the BC will become thin at first, and then break down into pieces.

Another strength of logistic model lies in that it ignored the hypothetical DP, making the result much more unbiased.

Sensitivity analysis of parameters

We perform a sensitivity analysis of Logistic equation's key parameters (R_m and lagging time) in the model to analyze their contributions and effect on the degradation rate.

Fig. 6 The effect of maximum velocity and lagging time on degradation

We transversed a wide range of values. Finally, we found that with the maximum velocity increase and lagging time descend, the degradation efficiency will improve. And when maximum velocity gets close to 0.015 and lagging time approach 570, the degradation performance changes sharply.

Michaelis-Menten equation

Michaelis-Menten equation without glucose inhibition

The reactions of degradation of cellulose into cellobiose follows the formula described below:

$$(C_{12} H_{22}O_{11})_{n} {\rightarrow}{ } \; n \times C_{12} H_{22}O_{11}$$

The corresponding equation by Michaelis-Menten equation are established as follows:

$$ \begin{array}{l} \frac{d[(C_{12} H_{22}O_{11})_{n}]}{dt} = -\frac{Vmax_{1} \,\times\, [(C_{12} H_{22}O_{11})_{n}]}{km_{1} \,+\, [(C_{12} H_{22}O_{11})_{n}]} \\ \frac{d[C_{12} H_{22}O_{11}]}{dt} = n\times \frac{Vmax_{1} \,\times\, [C_{12} H_{22}O_{11}]}{km_{1} \,+\, [C_{12} H_{22}O_{11}]} \\ \end{array} $$

Fig. 7 Degradation described by Michaelis-Menten equation

As we expected, cellulose will be degraded into cellobiose and follow the law of conservation of mass, which demonstrate the accuracy of our model.

Sensitivity analysis of parameters

We perform a sensitivity analysis of Michaelis-Menten equation's key parameters (V_max and k_m) in the model to analyze their contributions and effect on the degradation rate.

Fig. 8 Sensitivity analysis of V_max and k_m using contour line graph
(The contour lines represent the degradation efficiency)

Fig. 9 Sensitivity analysis of V_max and k_m using heat map

We transversed a wide range of values, and we found that with the maximum reaction velocity increase and k_m descent, the degradation efficiency will improve, which is coincided with biological pattern. And when maximum velocity gets close to 0.01, the degradation performance changes sharply.

Michaelis-Menten equation with glucose inhibition

Since the production of glucose will inhibit the activity of enzymatic activity of exo-1,4-D-glucanase and 1,4-D-glueanohydrolase. To make the model more accurate, we adjust our equations and involves the inhibitory process into it.

$$ \begin{array}{l} (C_{12} H_{22}O_{11})_{n} {\rightarrow}{ } \; n \times C_{12} H_{22}O_{11} \\ C_{12} H_{22}O_{11} \rightarrow 2 \times C_{6} H_{12}O_{6} \\ \end{array} $$

The reactions of degradation of cellulose into cellobiose described by Michaelis–Menten equation are as follows:

$$ \begin{array}{l} \frac{d[(C_{12} H_{22}O_{11})_{n}]}{dt} = -\frac{Vmax_{1} \,\times\, [(C_{12} H_{22}O_{11})_{n}]}{km_{1} \,+\, [(C_{12} H_{22}O_{11})_{n}]/k_{i}} \\ \frac{d[C_{12} H_{22}O_{11}]}{dt} = n\times \frac{Vmax_{1} \,\times\, [C_{12} H_{22}O_{11}]}{km_{1} \,+\, [C_{12} H_{22}O_{11}]/k_{i} } - \frac{Vmax_{2} \,\times\, [C_{6} H_{12}O_{6}]}{km_{2} \,+\, [C_{6} H_{12}O_{6}]}\\ \frac{d[C_{6} H_{12}O_{6}]}{dt} =\frac{Vmax_{2} \,\times\, [C_{6} H_{12}O_{6}]}{km_{2} \,+\, [C_{6} H_{12}O_{6}]} \\ \end{array} $$

Fig. 10 Degradation with inhibitor described by Michaelis-Menten equation

We found that when take glucose into consideration, the degradation rate of cellulose has inseparable bond between the inhibiting ability of glucose and activity of β-glucosidase. To make full use of experiment data, we tried to find relationship between mathematical model and experiment by script. However, we can get the simulation result by script, the parameter lacked biological meaning (the value is opposed to common sense). Based on the simulation result, the degradation rate of cellulose were decreased compared to without β-glucosidase. Meanwhile, we also found that the value of maximum velocity of degradation of cellobiose (V_max2) is lower than its original form, indicating that introducing the degradation of cellobiose is unnecessary to our project.

Sensitivity analysis of parameters

Since the complexity of reaction and limited to the experiment data, setting up a quantitative model is a thorny task, we perform a qualitative sensitivity analysis of the equation's key parameters in the model to analyze their contribution and effect on the degradation rate. We selected two important parameters, which are V_max1 and k_i and tries various combinations.

Fig. 11 Various parameter combinations analysis

We tried various parameter combinations. Interestingly, with the maximum reaction velocity and k_i descent, the degradation efficiency will improve. We speculate that this phenomenon is partially resulted from the cooperative effects of Vmax₂ and k_i. When the value of Vmax₁ is large, even the very tiny k_i will influence the degradation rate significantly; thus, slowing down the overall reaction speed. While k_i increases, the production of glucose decreases and cellobiose increases sharply, which fits into our biological routine.

Fig. 12 Combinational sensitivity analysis of β-glucosidase and Cex/CenA

Fig. 13 Combinational sensitivity analysis of inhibitor and β-glucosidase

Furthermore, we explored the effect of activity of enzyme degradation efficiency of cellulose. Since value of K_cat / K_m can evaluate the catalytic efficiency of enzyme and the concentration of enzyme is abundant in our project, we calculated the Vmax/Km as the evaluation index of activity of enzyme, based on the fact that V_max = K_cat × [E], where [E] is the concentration of enzyme.

Focusing on the Figure 13, we can indicate that the activity of glucose plays a key role in degradation of cellulose. The effect of the increasing of activity of Cex/CenA on degradation efficiency is weaker than that of β-glucosidase. Meanwhile, the activity of inhibitor can faintly perturb the degradation efficiency compared with β-glucosidase.

In summary, the cellulose will be degraded as expected under the weak activity of β-glucosidase and inhibitor, based on qualitative analysis. Actually, this situation is irrational in metabolism. From the above results we know that the degradation of cellulose to cellobiose has a higher efficiency compared to the degradation to glucose.

To model AAV6-bFGF 's effect on wound healing, we simplified and applied the multiscale hybrid mathematical model of epidermal-dermal interactions during skin wound healing ⁵. As scars typically have highly-parallel collagen ⁶ while collagen architecture of normal skin is intricate and weave-like ⁷ and direct contact between keratinocytes and fibroblasts would have a bad result on other cells, Epidermal and dermal interact with each other via signaling growth factors. During wound healing, ECM production and fibroblast proliferation are known to be regulated by multiple signaling factors. We divide them into two groups based on their effect on fibroblasts and excessive extracellular matrix (ECM). While Activator promotes fibroblast proliferation and ECM production and Inhibitor is counter-productive. For example, TGFβ1 is an activator and TGFβ3 shows downregulating effects thus it is an inhibitor. Signaling factors belonging to canonical WNT8 and PDGF pathways also have influence on fibroblast activity and can serve as activators and inhibitors in the wound healing model ⁴. As healing progresses, the density of fibroblasts and ECM are increasing ⁹, we expected to model the process through the change of the average density of ECM over time.

Fig. 14 Wound healing model

Our modified virus AAV6-bFGF ¹ can regulate the synthesis and deposition of various extracellular matrix components, increase the migration of keratinocytes and promote the migration of fibroblasts, thus accelerating the formation and re-epithelialization of granulation tissue. As Our team constructed the adeno-associated virus vectors containing effective fragments of AAV6-bFGF based on EGF and LL-37, our BC tissue engineering scaffold has the ability to assist in healing. We quantified its proliferative effect on fibroblasts and wrote it into the set of ordinary differential equations.

Here are the abbreviation table:

Model Assumption
(1) When healing starts, the wound contains no blood vessels; only a fibrin clot, cells and chemicals.
(2) The random motion of chemicals and cells in the wound area have no difference on their total quantity in the wound area, and then their average density in the model.
(3) The modeling domain is separated into epidermis (E) (light blue) and dermis (D) (grey) by a dynamic interface (Ω) to mimic the basement membrane. The dynamic interface has no effect on the amount of ECM, Fibroblasts and chemicals.
(4) Both dermal and epidermal signaling factors are indispensable in sustaining steady-state ECM levels in normal skin and regulating new ECM deposition after wounding. Both basal epidermal keratinocytes and dermal fibroblasts produce A and I. Immune cells can also produce A⁵.

Results

$$\begin{array}{l} \frac{d[F]}{d t}=\frac{c_{F} F}{1+\left(\frac{F}{g m_{F}}\right)^{n} F}\left[b_{F}+\frac{a_{F} p_{2}^{n_{p2F} }}{1+\left(\frac{p_{2}}{g m_{p 2 F}}\right)^{n_{p 2 F}}}\right]-d_{F} F\\ \frac{d[C]}{d t}=c_{c} F\left[b_{c p 1}-\frac{a_{c p 1} p 1^{n_{p 1}}}{1+\left(\frac{p 1}{g m_{p 1 c}}\right)^{n_{p 1 c}}}\right]\left[b_{c p 2}+\frac{a_{c p 2} p 2^{n_{p 2}}}{1+\left(\frac{p p^{2}}{g m_{p 2 c}}\right)^{n_{p 2 c}}}\right]-d_{c} C\\ \frac{d\left[p_{1}\right]}{d t}=c_{F p 1}[b_{p 1}+\frac{a_{p 1} C^{n_{p 1}}}{1+(\frac{C}{g m_{p 1}})^{n_{p 1}}}]+E_{p 1}+c_{M P 1} M-d_{p 1} p 1\\ \frac{d[B]}{d t}=\frac{c_{F p 2} F}{[b_{p 2}+\frac{a_{p 2} C^{n_{p2}} }{1+(\frac{C}{g m_{p 2}})^{n_{p 2}}}]}+E_{p 2}+c_{M p 2} M+I M \frac{\left(p_{2}-p_{l i m}\right)^{n_{i m}}}{1+\left(p_{2}-p_{l i m}\right)^{n_{i m}}}-d_{p_{2}} p_{2}\\ \frac{d[M]}{d t}=D_{M}M_{0}-d_{M}M \end{array} $$

Using the specific set of parameters that we sourced from literature⁵, we modeled the concentration change of ECM in the case of normal wound and in the case of wound with AAV6-bFGF in our BC membrane.

Fig.15 Average density of ECM

The accumulation of ECM reached stability over time (Fig 15). The result shows that our BC membrane assists in healing which means it contributes to the rate-increasing of wound healing.