Protein Estimation

We tried to estimate the amount of IPTG that would be required for the optimal protein production so as to predict and thereby help with the experimental design.

We also predicted the max amount of protein that a single E.coli cell would be able to produce and further went on to verify our predictions with experimental results. To get IPTG vs Protein output and Time vs Max Protein output curve, we tried modeling every metabolic process that takes place inside the cell after IPTG induction that is responsible for the production of our enzyme.

The gene of our interest is expressed in the BL21 DE(3) strain of E.coli bacteria, whose expression will be under the inducible promoter. The inducer here is IPTG (Isopropyl β-d-1-thiogalactopyranoside). The T7 RNA polymerase which binds to the promoter of our gene of interest and helps in transcription is further under the control of the Lac operon. The IPTG thus have to bind with both:

• Lac repressor produced by LacI gene in genomic DNA (to enable the transcription of T7 RNA polymerase) and
• Lac repressor produced by LacI gene in plasmid DNA so as to free the site at which T7 RNA Polymerase binds and transcribes the gene of interest.

The metabolic processes for Enzyme production are described in terms of the following Ordinary differential equations (ODEs):^[9]

For Genomic Part of E-coli


• Transcription of LacI mRNA: \[\begin{equation} \phi\stackrel{K_{tcI}}{\longrightarrow}mRNA_{LacI}
  \end{equation}\] Decay of LacI mRNA: \[\begin{equation} mRNA_{LacI}\stackrel{K_{dmI}}{\longrightarrow}\phi
  \end{equation}\]
\[\begin{equation} \frac{d[mRNA_{LacI}]}{dt}=K_{tcI}-K_{dmI}[mRNA_{LacI}]\tag{1} \end{equation}\]
• Translation of LacI: \[\begin{equation} mRNA_{LacI}\stackrel{K_{tl}}{\longrightarrow}mRNA_{LacI}+LacI
  \end{equation}\] Decay of LacI: \[\begin{equation} LacI\stackrel{K_{dpI}}{\longrightarrow}\phi \end{equation}\] \[\begin{equation} \frac{d[LacI]}{dt}=K_{tl}[mRNA_{LacI}]-K_{dpI}[LacI]\tag{2} \end{equation}\]
• Formation of LacI and IPTG complex: \[\begin{equation} LacI+IPTG_{in} \underset{K_{bI}}{\stackrel{K_{fI}}{\rightleftharpoons}} LacI-IPTG_{in} \end{equation}\] \[\begin{equation} \frac{d[LacI-IPTG_{in}]}{dt}=K_{fI}[LacI][IPTG_{in}]-K_{bI}[LacI-IPTG_{in}]\tag{3} \end{equation}\]
• Passive diffusion of IPTG across the cell membrane of E. coli: \[\begin{equation} IPTG_{ex} \underset{K_{pdiff}}{\stackrel{K_{pdiff}}{\rightleftharpoons}} IPTG_{in} \end{equation}\] \[\begin{equation} \frac{d[IPTG_{ex}]}{dt}=K_{pdiff}([IPTG_{ex}]-[IPTG_{in}])+K_{fdiff}([LacY-IPTG_{ex}])\tag{4} \end{equation}\]
• Transcription of LacY mRNA: \[\begin{equation} O\stackrel{K}\longrightarrow O+mRNA_{LacY} \end{equation}\] Decay of LacY mRNA: \[\begin{equation} mRNA_{LacY}\stackrel{K_{dmY}}{\longrightarrow}\phi \end{equation}\] \[\begin{equation} \frac{d[mRNA_{LacY}]}{dt}=K_{tcY}[O_T]-K_{dmY}[mRNA_{LacY}]\tag{5} \end{equation}\]
• Translation of LacY: \[\begin{equation} mRNA_{LacY}\stackrel{K_{tlY}}{\longrightarrow}RNA_{LacY}+LacY \end{equation}\] Decay of LacY: \[\begin{equation} LacY\stackrel{K_{dY}}{\longrightarrow}\phi\ \end{equation}\] \begin{equation} \frac{d[LacY]}{dt}=K_{tlY}[mRNA_{LacY}]+(K_{pdiff}+K_{bY})[LacY-IPTG_{ex}]-K_{fY}[LacY][IPTG_{ex}]-K_{dY}[LacY]\tag{6} \end{equation} Formation of IPTG-LacY complex and facilitated diffusion \[\begin{equation} LacY+IPTG_{ex}\underset{K_{bY}}{\stackrel{K_{fY}}{\rightleftharpoons}}LacY-IPTG\stackrel{K_{fdiff}}{\longrightarrow}LacY+IPTG_{in} \end{equation}\]
• Formation of \(LacY-IPTG_{ex}\) complex: \[\begin{equation} LacY+IPTG_{ex}\underset{K_{bY}}{\stackrel{K_{fY}}{\rightleftharpoons}}LacY-IPTG_{ex} \end{equation}\]
\[\begin{equation} \frac{d[LacY-IPTG_{ex}]}{dt}=-(K_{pdiff}+K_{bY})[LacY-IPTG]_{ex}+K_{fY}[LacY][IPTG]-K_{dY\_IPTG}[LacY-IPTG_{ex}]\tag{7} \end{equation}\]
• Free LacI present (that is, not bounded to IPTG), which will prevent the transcription of T7 RNA polymerase
  
\[\begin{equation} \frac{d[LacI]_F}{dt}=(K_{tl}[mRNA_{LacI}]-K_{dpI}[LacI])-(K_{fI}[LacI][IPTG_{in}]-K_{bI}[LacI-IPTG_{in}])\tag{8} \end{equation}\]
From here, \([IPTG_{in}]\) will be replaced by \([IPTG_{Pin}]\) ; intracellular IPTG due to facilitated diffusion by LacY produced by the plasmid of E.coli due to its high copy number. The [IPTG] diffused by this virtue can bind to LacI-repressor produced by the genomic as well.


• Production of T7 RNA polymerase mRNA: \[\begin{equation} \phi\stackrel{K_{tcT7}}{\longrightarrow}mRNA_{T7}\ \end{equation}\] Decay of T7 RNA polymerase mRNA: \[\begin{equation} mRNA_{T7}\stackrel{K_{dmT7}}{\longrightarrow}\phi\ \end{equation}\] Binding of free LacI and \(mRNA_{T7}\): \[\begin{equation} LacI_{F}+mRNA_{T7}\stackrel{K_{aff}}{\longrightarrow}\phi \end{equation}\]
\[\begin{equation} \frac{d[mRNA_{T7}]}{dt}=K_{tcT7}-K_{dmT7}[mRNA_{T7}]-K_{aff}[LacI]_F \tag{9} \end{equation}\]
• T7 RNA Polymerase production : \[\begin{equation} mRNA_{T7}\longrightarrow{K_{tlT7}}mRNA_{T7}+T7\_RNA\_pol \end{equation}\] \[\begin{equation} T7\_RNA\_pol\stackrel{K_{dpT7}}{\longrightarrow}\phi \end{equation}\]
\[\begin{equation} \frac{d[T7\_RNA\_pol]}{dt}=K_{tlT7}[mRNA_{T7}]-K_{dpT7}[T7\_RNA\_pol] \tag{10} \end{equation}\] For the plasmid part of the E.coli:


• \[\begin{equation} \frac{d[mRNA_{PLacI}]}{dt}=K_{PtcI}-K_{PdmI}[mRNA_{PLacI}]\cdot (Copy\quad number) \tag{11} \end{equation}\]
• \[\begin{equation} \frac{d[LacI_P]}{dt}=K_{Ptl}[mRNA_{PLacI}]-K_{PdpI}[LacI] \tag{12} \end{equation}\]
• \[\begin{equation} \frac{d[LacI-IPTG_{in}]_P}{dt}=K_{PfI}[LacI_P][IPTG_{Pin}]-K_{PbI}[LacI-IPTG_{in}]_P \tag{13} \end{equation}\]
• \[\begin{equation} \frac{d[IPTG_{Pin}]}{dt}=K_{Ppdiff}([IPTG]_{ex}-[IPTG_{Pin}])+K_{Pfdiff}([LacY_P-IPTG_{Pex}]) \tag{14} \end{equation}\]
• \[\begin{equation} \frac{d[mRNA_{LacY}]_P}{dt}=K_{PtcY}[O_T]-K_{PdmY}[mRNA_{LacY}]_P \tag{15} \end{equation}\]
• \[\begin{equation} \frac{d[LacY]_P}{dt}=K_{PtlY}[mRNA_{LacY}]_P+(K_{Ppdiff}+K_{PbY})[LacY\_IPTG_{ex}]_P-K_{PfY}[LacY][IPTG_{ex}]-K_{PdY}[LacY] \tag{16} \end{equation}\]
• \[\begin{equation} \frac{d[LacY\_IPTG_{ex}]_P}{dt}=-(K_{Ppdiff}+K_{PbY})[LacY-IPTG]_{Pex}+K_{PfY}[LacY]_P[IPTG_{ex}]-K_{PdY\_IPTG}[LacY\_IPTG_{ex}]_P \tag{17} \end{equation}\]
• \[\begin{equation} \frac{d[LacI]_{PF}}{dt}=(K_{Ptl}[mRNA_{LacI}]_P-K_{PdpI}[LacI]_P)-(K_{PfI}[LacI]_P[IPTG_{Pin}]-K_{PbI}[LacI-IPTG_{in}]_P) \tag{18} \end{equation}\] Binding of T7 RNA Polymerase: \[\begin{equation} RNA\_pol\underset{K_{ub}}{\stackrel{K_{b}}{\rightleftharpoons}}RNA\_pol_{B} \end{equation}\] \[\begin{equation} \frac{d[T7\_RP_B]}{dt}=K_b[T7\_RNA\_pol]-K_{ub}[T7\_RP_B]-K_{aff}[LacI]_F \tag{19} \end{equation}\] Decay of mRNA Chitinase: \[\begin{equation} mRNA_{Chitinase}{\stackrel{K_{dmChitinase}}{\longrightarrow}}\phi \end{equation}\] \[\begin{equation} \frac{d[mRNA_{Chitinase}]}{dt}=K_{leaky}+\frac{K_1[T7\_RP_B]^n}{[K_m]^n+[T7\_RP_B]^n}-K_{dmChitinase}[mRNA_{Chitinase}] \tag{20} \end{equation}\] Chitinase production: \[\begin{equation} mRNA_{Chitinase}{\stackrel{K_{tlChitinase}}{\longrightarrow}}mRNA_{Chitinase}+Chitinase \end{equation}\] Decay of the chitinase protein: \[\begin{equation} Chitinase{\stackrel{K_{dChitinase}}{\longrightarrow}}\phi \end{equation}\] \[\begin{equation} \frac{d[Chitinase]}{dt}=K_{tlChitinase}[mRNA_{Chitinase}]-K_{dChitinase}[Chitinase]\tag{21} \end{equation}\]

Symbol	Description
[\(mRNA_{LacI}\)]	mRNA of LacI
[LacI]	LacI
[\(IPTG_{in}\)]	Intracellular IPTG
[\(IPTG_{ex}\)]	Extracellular IPTG
[\(mRNA_{LacY}\)]	mRNA of LacY
[LacY]	LacY
[\(LacY - IPTG_{ex}\)]	\(LacY - IPTG_{extracellular}\) complex
[\(LacI - IPTG_{in}\)]	\(LacI - IPTG_{intracellular}\) complex
[\(LacI_{F}\)]	Free (unbounded) LacI
[\(mRNA_{\text{T7 RNA POL}}\)]	mRNA of T7 RNA polymerase
[T7 RNA POL]	T7 RNA polymerase
[T7 \(RP_{B}\)]	T7 RNA polymerase bound to promoter region
[\(mRNA_{\text{chitinase}}\)]	mRNA of Chitinase
[Chitinase]	Chitinase

Note: The elements for the plasmid part are represented with the same Symbols but have an additional “P” added to them to signify plasmid.

Symbol	Description	Value
K_{tcI}	Rate constant for transcription of LacI	0.23nMmin^(-1)[9]
K_{PtcI}
K_{dmI}	Degradation constant of LacI mRNA	0.462 min^(-1)[9]
K_{PdmI}
K_{tl}	Rate constant for translation of LacI	15 min^(-1)[9]
K_{Ptl}
K_{dpI}	Degradation constant for LacI	0.2 min^(-1)[9]
K_{PdpI}
K_{pdiff}	Rate constant for passive diffusion of IPTG	0.92 min^(-1)[9]
K_{Ppdiff}
K_{fdiff}	Rate constant of facilitated diffusion	6*10^4 min^(-1)[9]
K_{Pfdiff}
K_{fY}	Forward reaction rate constant for formation of LacY - IPTG complex	0.12nM^(-1)min^(-1)[9]
K_{PfY}
K_{bY}	Backward reaction rate constant for the formation of LacY - IPTG complex	0.1 min^(-1)[9]
K_{PbY}
K_{dYI}	Degradation constant for LacY - IPTG complex	0.2 min^(-1)[9]
K_{PdYI}
K_{fI}	Forward reaction rate constant for formation of LacY - IPTG(intracellular) complex	0.12 nM^(-1)min^(-1)[9]
K_{PfI}
K_{bI}	Backward reaction rate constant for formation of LacI - IPTG(intracellular) complex	12 min^(-1)[9]
K_{PbI}
K_{tcY}	Rate constant for transcription of LacY	0.5nMmin^(-1)[9]
K_{PtcY}
K_{dmY}	Degradation constant of LacY mRNA	0.462 min^(-1)[9]
K_{PdmY}
K_{tlY}	Rate constant of translation of LacY	30 min^(-1)[9]
K_{PtlY}
K_{dY}	Degradation constant for LacY	0.2 min^(-1)[9]
K_{PdY}
K_{tcT7}	Rate constant of transcription of T7 RNA Pol	0.5 nMmin^(-1)[9]
K_{dmT7}	Degradation constant of T7 RNA Pol mRNA	0.462 min^(-1)[9]
K_{aff}	Affinity constant of LacI to the operator region	0.2 min^(-1)[9]
K_{tlT7}	Rate constant of translation of T7 RNA Pol	6.116 min^(-1)[9]
K_{dpT7}	Degradation constant of T7 RNA Pol	0.462 min^(-1)[9]
K_{b}	Binding constant of T7 RNA to Promoter	0.22 min^(-1)[9]
K_{ub}	Unbinding constant	28.8 min^(-1)[9]
K_{leaky}	Rate constant for leaky expression	0.01 nMmin^(-1)[9]
K_m	Michaelis Menten constant	3*10^5 nM[9]
K_1	Transcription Rate constant of Chitinase mRNA	2.3658 * 10^3 nMmin^(-1)[9]
K_{dm_chitinase}	Degradation Rate constant for Chitinase mRNA	0.462min^-1[9]
K_{tl_chitinase}	Translation Rate constant for Chitinase	10 min^-1[9]
K_{dp_chitinase}	Degradation constant for chitinase	0.2 min^-1[9]

Through our model, we were able to predict that at the concentration of 0.1 - 1 mM IPTG, we can obtain optimal protein production. This prediction helped us in the design of experiments concerning the expression of the protein. It must be noted that these predictions are well in line with the standard procedures that are being used for IPTG induction.

Our other area of interest was predicting the total amount of enzyme that could be produced by a single E Coli: Through experimental data, we got to know that 2 litres of culture produce about 20 mg of protein. Assuming there are 10^10 bacteria / L [10] of culture (general assumption) we were able to calculate the amount of protein produced by a single E.coli to be ~ 256000 nM. These experimental results are very close to our prediction which is 255853.577 nM/ cell.

Predicting enzyme activity

Since our chimeric Chitinase comprises domains from two different chitinases, with both of them having a binding domain, we could, in theory, predict the activity of our chitinase by analyzing the binding of the substrate with individual domains and further coupling the activity to get a combined activity.^[11]

\[\begin{equation} K_{D1}=\frac{[E][S]}{[ES]}\implies [ES]=\frac{[E][S]}{K_{D1}} \tag{1} \end{equation}\] \[\begin{equation} K_{D2}=\frac{[ES][S]}{[ESS]}\implies [ESS]=\frac{[ES][S]}{K_{D2}}=\frac{[E][S]^{2}}{K_{D1}K_{D2}} (Using\quad (1)) \end{equation}\] Relative to the Total Enzyme Concentration, \(E_T\) \[\begin{equation} \frac{[ES]}{[E_{T}]}=\frac{[ES]}{[E]+[ES]+[ESS]}=\frac{\frac{[E][S]}{K_{D1}}}{[E]+\frac{[E][S]}{K_{D1}}+\frac{[E][S]^{2}}{K_{D1}K_{D2}}}=\frac{\frac{[S]}{K_{D1}}}{1+\frac{[S]}{K_{D1}}+\frac{[S]^2}{K_{D1}K_{D2}}} \end{equation}\] \[\begin{equation} \frac{[ESS]}{[E_T]}=\frac{[ESS]}{[E]+[ES]+[ESS]}=\frac{\frac{[E][S]^2}{K_{D1}K_{D2}}}{[E]+\frac{[E][S]}{K_{D1}}+\frac{[E][S]^2}{K_{D1}K_{D2}}}=\frac{\frac{[S]^2}{K_{D1}K_{D2}}}{1+\frac{[S]}{K_{D1}}+\frac{[S]^2}{K_{D1}K_{D2}}} \end{equation}\] The total velocity of this system is given by:
\[\begin{equation} v=\frac{V_{max1}\frac{[S]}{K_{D1}}+V_{max2}\frac{[S]^2}{K_{D1}K_{D2}}}{1+\frac{[S]}{K_{D1}}+\frac{[S]^2}{K_{D1}K_{D2}}} \end{equation}\] Where,
\(K_{D1}\)= Dissociation constant for chitinase of Serratia marcescens
\(K_{D2}\)= Dissociation constant for chitinase of Amycolatopsis orientalis
\(V_{max1}=K_{cat1}[E_T] and V_{max2}=K_{cat2}[E_T]\)

We here analyse the activity of bacterial Chitinase combo 2 consisting of domains from two chitinases - Serratia marcescens chitinase and Streptomyces orientalis chitinase. Here instead of Streptomyces orientalis we have used proxy species of the same genus Streptomyces griseus as the literature data for Streptomyces orientalis was unavailable. The dissociation constants and Vmax for both chitinases were determined from the literature survey.^[12][13]

K_{D1}	3.36*10^(-5)
K_{D2}	4.43*10^(-6)
Vmax1	0.18mmol/min
Vmax2	2.4mmol/min

The Vmax for chitinase was found to be 2.3739 mmol/min, which gives us an idea of how adding domains from high activity enzymes could in theory improve the efficiency of low activity enzymes.

Further plotting Lineweaver Burk Plot:

Nanoencapsulation

For the delivery of our chimeric chitinase into the human system, we are looking forward to proposing/ using nanoparticles (specifically PLGA) that are suitable due to their ability for controlled drug release, improved therapeutic efficiency, and reduction in the dose of drug administered.

Upon doing assays using our engineered chimeric protein, we have found that our protein is stable at room temperature and shows promising activity.Thus, it further strengthens our proposal to use PLGA as a delivery molecule to the human body.

PLGA - Poly(lactic-co-glycolic acid) is a synthetic copolymer made up of Lactic Acid (LA) and glycolic Acid(GA) and is extensively used for controlled drug delivery(approved by FDA). In an aqueous environment, the chain undergoes a hydrolytic attack on its ester bonds and degrades into LA and GA which are endogenous compounds normally metabolized through the Krebs cycle. By changing the LA/GA ratio and molecular weight in the polymer we could easily control the degradation time of the molecule. The degradation of the PLGA microsphere typically starts at the center of the sphere and the degradation front moves outwards.

Here we are trying to model and predict the degradation kinetics of PLGA (LA/GA ratio 50:50 and Mw 41.9 KDa in human body which helps us understanding the controlled drug release in human body.

Protein release from PLGA Microsphere(MS) is an autocatalytic polymer degradation mechanism, involving 3 prominent steps:^[14]

• Water intruding in the device through its micro/macropores, and subsequent occurring of matrix plasticization
• Hydrolytically initiation of PLGA degradation at chain backbone and production of acidic by-products
• Produced acids further catalyze polymer degradation

Assumptions made

• Spherical symmetry
• Invariant geometry during protein unloading

PLGA degradation governs protein solubilization and release. PLGA degradation is promoted in the MS center due to the polymer autocatalytic degradation mechanism, causing a decrease in pH and enhancing the degradation process.

\[\begin{equation} \rho=\frac{r}{R_{MS}} \tag{1} \end{equation}\] Where,
\(\rho\)= Non-dimensional results
RMS=Average radius of Microsphere
r= Generic radius within Microsphere
Neglecting the initial hydration phase,
\[\begin{equation} A+P \stackrel{k_1,k_{-1}}{\longleftarrow} A \cdot P \stackrel{k_2}{\longrightarrow} A+S \tag{2} \end{equation}\]
Where,
A= Soluble acid (carbonic acid in present blood)
P= Non-degraded polymer acting as substrate
\(A\cdot P\)= Partially degraded, insoluble polymer
\(k_1\)= Rate constant for forward reaction
\(k_{-1}\)= Rate constant for backward reaction
\(k_2\)= Rate constant for final degradation
Equation (2) can be considered as a Michaelis-Menten equation. Approximating the quasi steady state, we have,
\[\begin{equation} k_1[A][P]=(k_{-1}+k_2)[A\cdot P] \tag{3} \end{equation}\] \[\begin{equation} [A\cdot P]=\frac{k_1[A]_0[P]}{k_1[P]+k_{-1}+k_2} \tag{4} \end{equation}\] Where,
\([A]_0\)= Initial acid concentration, and \([A]+[P]=[A]_0\)
The rate of soluble product generation,
\[\begin{equation} \frac{d[S]}{dt}= k_2[A\cdot P]=\frac{k_1k_2[A]_0[P]}{k_1[P]+k_{-1}+k_2} \tag{5} \end{equation}\]
\[\begin{equation} \frac{d[S]}{dt}=\frac{k_0[P]}{K_M+[P]} \tag{6} \end{equation}\] Where,
\(K_M=\frac{(k_{-1}+k_2)}{k_1}\), estimates the polymer-acid affinity, low KM indicates high affinity
\(R_0=k_2\cdot A_0\)
\[\begin{equation} [S]=\frac{m_S}{V_{MS}MW_S} \tag{7} \end{equation}\] Where,
\(m_S\)= Mass of soluble products
\(MW_S\)= Molecular weight of soluble products
\(V_{MS}\)= Volume of a single MS
\[\begin{equation} \frac{d[S]}{dt}=\frac{1}{V_{MS}MW_S}\frac{dm_S}{dt} \tag{8} \end{equation}\] Substituting (8) in equation (6), \[\begin{equation} \frac{dm_S}{dt}=V_{MS}MW_S\frac{R_0[P]}{K_M+[P]} \tag{9} \end{equation}\]
\[\begin{equation} [P]=\frac{m_P}{V_{MS}MW_P} \tag{10} \end{equation}\] Where,
\(m_P\)= Time-dependent mass of PLGA
\(MW_P\) = Number-average molecular weight of PLGA
Substituting (10) in equation (9), \[\begin{equation} \frac{dm_S}{dt}=V_{MS}MW_S\frac{R_0m_P}{K_MV_{MS}MW_P+m_P} \tag{11} \end{equation}\]
\[\begin{equation} \mu _S=\frac{m_S}{m_{MS,0}} \tag{12} \end{equation}\]
\[\begin{equation} \mu _P=\frac{m_p}{m_{MS,0}} \tag{13} \end{equation}\]
\(\mu_P + \mu_S=1\) \[\begin{equation} \omega _P=\frac{MW_P}{MW_{P,0}} \tag{14} \end{equation}\]
Where,
\(m_{MS,0}\)=Mass of single microspore at time 0
\(MW_{P,0}\)= PLGA molecular weight at time 0
\(\mu_P\)= Non-dimensional mass of residual polymer
\(\mu_S\)= Non-dimensional mass of soluble products
\(\omega _P\)= Non-dimensional polymer weight normalized with respect to its initial value
\[\begin{equation} \frac{d\mu_S}{dt}=\frac{B_0(1-\mu _S)}{\psi _{\omega_{P}}+1-\mu _S} \tag{15} \end{equation}\]
\[\begin{equation} B_0=\frac{V_{MS}MW_SR_0}{m_{MS,0}} \tag{16} \end{equation}\]
\[\begin{equation} \psi=\frac{K_MV_{MS}MW_{P,0}}{m_{MS,0}} \tag{17} \end{equation}\]
\[\begin{equation} \omega _P=e^{(-k_{deg}t)} \tag{18} \end{equation}\] Where,
\(k_{deg}\)= Degradation constant of the polymer
Polymer loss (i.e., generation of soluble acids) takes place after an induction time, \(t_{ind}\),
Thus, equation (15) can be modified by including a heaviside function, as follows,
\[\begin{equation} \frac{d\mu_S}{dt}=\frac{B_0(1-\mu _S)}{\psi _{\omega_{P}}+1-\mu _S}H(t-t_{ind}) \tag{19} \end{equation}\]
To account for the continuous (non-discrete) transition of µS in time, Heaviside function has been mollified as shown in the following Equation \[\begin{equation} \frac{d\mu_S}{dt}=\frac{1}{2}\frac{B_0(1-\mu _S)}{\psi _{\omega_{P}}+1-\mu _S}[tanh[\alpha (t-t_{ind})]+1] \tag{20} \end{equation}\]
given the initial condition of \[\begin{equation} µ_S(0)=0 \tag{21} \end{equation}\]
Assuming density of PLGA is constant during degradation, equation (14) can be written as,
\[\begin{equation} \mu _P=\frac{V_P}{V_{MS}}=1-R^3_f(t) \tag{22} \end{equation}\]
\[\begin{equation} R_f(t)=\sqrt[3]{\mu _S} \tag{23} \end{equation}\]

Note: \(B_0,t_{ind}, \psi\) are all adjustable parameters.

This model predicts that our PLGA microsphere(Mw = 41.9 KDa) will undergo full disintegration in about 60 days (2 months) in physiological conditions. The rate of protein release will be in proportion with the decay rate of the microsphere and hence the dynamics of protein released could be analyzed using this model.

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