Team:IISER Kolkata/Model
Math modelling
Overview
Phagocytosis
Modelling of Phagocytosis of the GMO by the Macrophage:
Our system has been modelled in vitro. The GMO will be inserted in a Macrophage culture and will enter the macrophage by the process of phagocytosis. We model the dynamics of the GMO entering the Macrophage using a 3 compartment system A,B and C.
A : The GMOs adherent to the cell membrane of the Macrophage
B : The GMOs that has been phagocytosed
C : The phagocytosed GMOs that fuse with the lysosome (death of the GMO)
$$ \stackrel{R_a}\rightarrow A \stackrel{k_i}\rightarrow B \stackrel{k_f}\rightarrow C $$
The GMO present outside the Macrophage first adheres to the surface of the Macrophage and then the Macrophage engulfs these adherent bacteria (phagocytosis). The GMOs are now inside phagosomes. These phagosomes eventually fuse with the lysosome causing the death of our GMO. Our GMO will be active while it is in compartment B therefore we focus on the population of our GMO in this compartment.
Let \(B_a\) , \(B_b\) and \(B_c\) be the population of our GMO in the compartments A ,B and C respectively. We assume first order kinetics, therefore we can write the following differential equations:
$$\frac{dB_a}{dt} \space = \space -K_i B_a \space + \space R_a $$ $$\frac{dB_b}{dt} \space = \space K_i B_a \space - \space K_f B_b$$ $$\frac{dB_c}{dt} \space = \space K_f B_b$$
Where, \( \space R_a\) is the rate of adherence of our GMO to the surface of the Macrophage \(K_i\) is the rate of intake of GMO into the macrophage \(K_f\) is the rate of uptake of the GMO by lysosomes (death of the GMO)
For Peritonial Macrophages we have the values:\(^{[1]}\)
\( R_a \space = \space 0.011 \space \space No. \space of \space GMO \space per \space cell/min \) ( \(t<\) \(15 min\))
\( 0.002 \space \space No. \space of \space GMO \space per \space cell/min \) ( \(t>\) \(15 min\) )
\(K_i \space = \space 0.047 \space \space min^{-1} \)
\(K_f \space = \space 0.109 \space \space min^{-1} \)
Using these values we get the following graph from our simulation:
Results
- The system achieves steady state in 120 to 150 minutes in compartment A (\(4.26 \space \) GMOs\( \space per \space macrophage\)) and compartment B (\(1.82 \space \) GMOs\( \space per \space macrophage\))
- In the steady state, there are nearly 4 active GMOs per macrophage. This data will be utilised in the Iron regulation modelling.
References
- Bizal, C. L., Butler, J. P., Feldman, H. A. and Valberg, P. A. (1991), Kinetics of Phagocytosis and Phagosome‐Lysosome Fusion in Hamster Lung and Peritoneal Macrophages. J Leukoc Biol, 50: 229-239. doi:10.1002/jlb.50.3.229
Nitric oxide sensor
The most fundamental part of the project is the Nitric Oxide (\(NO\)) sensor. The genetic circuit is designed to operate in a narrow band of \(NO\) concentration. Ideally, the circuit should be active between 2 concentrations \(C_1\) and \(C_2\). \(C_1\) is the concentration slightly greater than the normal Nitric Oxide concentration of an uninfected Macrophage (which is around \(1 \space \mu M \space ^{[1]}\)). \(C_2\) is the concentration slightly less than the concentration of Nitric Oxide inside an infected Macrophage (which is around \(7-28 \space \mu M \space ^{[1]}\)). It is important to test the circuit using a model in order to check for viability and to give useful insights to the wet lab team.
Main Goals of this Model
- To see the activity of the genectic circuit under various Nitric Oxide concentrations/ To test the sensitivity of the Nitric Oxide Sensor
- To identify the parameters that can be tuned to get the desired Nitric Oxide sensing range
- To test the response time of our circuit to the environment of the Leishmania infected Macrophage
- To suggest improvements that can be made to our circuit to create a better sensor.
To model the Genetic Circuit, we first consider the various reactions and interactions involved:
\(NorR\) and \(NsrR\) are protiens that sense Nitric oxide. Nitric oxide binds with \(NorR\) and \(NsrR\) to form complexes. The \(NorR-NO\) complex further acts as an activator for the transcription of \(TetR\). \(NsrR\) and \(TetR\) bind to the \(NsrR-binding-site\) and \(TetR-binding-site\) respectively at the operator region of our Aerobactin producing part. Hence, repressing the production of Aerobactin
$$ NsrR + NO \underset{k_{b_1}}{\stackrel{k_{f_1}}{\rightleftharpoons}} NsrR-(NO) $$ $$ NsrR-(NO) + NO \underset{k_{b_2}}{\stackrel{k_{f_2}}{\rightleftharpoons}} NsrR-(NO)_2 $$ $$ NorR + NO \underset{k_{b_3}}{\stackrel{k_{f_3}}{\rightleftharpoons}} NorR-NO $$
The kinetic equations for the above reactions can be written as follows: $$r_1 \space = \space k_{f_1}[NsrR][NO] - k_{b_1}[NsrR-(NO)] $$ $$r_2 \space = \space k_{f_2}[NsrR-(NO)][NO] - k_{b_2}[NsrR-(NO)_2] $$ $$r_3 \space = \space k_{f_3}[NorR][NO] - k_{b_3}[NorR-(NO)]$$
We can express the rate of change of concentrations as follows: $$\frac{d[NsrR]}{dt} \space = \space \beta_{NsrR} \space - \space \alpha_{NsrR} [NsrR] - \space r_1$$ $$\frac{d[NorR]}{dt} \space = \space \beta_{NorR} \space - \space \alpha_{NorR} [NorR] - \space r_3$$ $$ \frac{d[TetR]}{dt} \space = \space \frac{\beta_{TetR}}{1 \space + \space \frac{K_{Tet}}{[NorR-(NO)]^2}} \space - \space \alpha_{TetR} [TetR]$$ $$\frac{d[NsrR-(NO)]}{dt} \space = \space r_1 - \space \alpha_{NsrR-(NO)} [NsrR-(NO)]$$ $$\frac{d[NsrR-(NO)_2]}{dt} \space = \space r_2 \space - \space \alpha_{NsrR-(NO)_2} [NsrR-(NO)_2]$$ $$\frac{d[NorR-(NO)]}{dt} \space = \space r_3 \space - \space \alpha_{NorR-(NO)} [NorR-(NO)]$$
The rate of production (\(r_{Ab}\)) of our Iron Chelator Aerobactin at any time instant can be expressed as: $$ \space r_{Ab} \space = \space \frac{\beta_{Ab}}{1 \space + \space \frac{[NsrR]^2}{K_{D_1}} \space + \space \frac{[TetR]^2}{K_{D_2}}}$$ Or the Relative rate of production (\(f_{Ab}\)) can be given by: $$ \space f_{Ab} \space = \space \frac{r_{Ab}}{\beta_{Ab}} \space = \space \frac{1}{1 \space + \space \frac{[NsrR]^2}{K_{D_1}} \space + \space \frac{[TetR]^2}{K_{D_2}}}$$
Parameters
| Parameter | Description | Value | Unit | Source |
|---|---|---|---|---|
| \(k_{f_1}\) | \(NsrR-(NO)\) production rate | \(271.2\) | \(\mu M ^{-1} \space min^{-1}\) | |
| \(k_{b_1}\) | \(NsrR-(NO)\) dissociation rate | \(100\) | \(min^{-1}\) | |
| \(k_{f_2}\) | \(NsrR-(NO)_2\) production rate | \(6.78\) | \(\mu M ^{-1} \space min^{-1}\) | |
| \(k_{b_2}\) | \(NsrR-(NO)_2\) dissociation rate | \(100\) | \(min^{-1}\) | |
| \(k_{f_3}\) | \(NorR-(NO)\) production rate | \(1.1183\) | \(\mu^{-1}min^{-1}\) | |
| \(k_{b_3}\) | \(NorR-(NO)\) dissociation rate | \(0.0363665\) | \(min^{-1}\) | |
| \(\alpha_{NsrR}\) | Degradation rate of \(NsrR\) | \(0.0015\) | \(min^{-1}\) | |
| \(\alpha_{NorR}\) | Degradation rate of \(NorR\) | \(89.159\) | \(min^{-1}\) | |
| \(\alpha_{TetR}\) | Degradation rate of \(TetR\) | \(0.0348\) | \(min^{-1}\) | |
| \(K_{Tet}\) | Activation rate of \(TetR\) | \(0.5×10^{-4} \) | \(\mu M^2\) | |
| \(\alpha_{NsrR-(NO)}\) | Degradation rate of \(NsrR-(NO)\) | \(1\) | \(min^{-1}\) | |
| \(\alpha_{NsrR-(NO)_2}\) | Degradation rate of \(NsrR-(NO)_2\) | \(1\) | \(min^{-1}\) | |
| \(\alpha_{NorR-(NO)}\) | Degradation rate of \(NorR-(NO)\) | \(0.0363665\) | \(min^{-1}\) |
Tunable Parameters
- The expression rates of \(NorR\), \(NsrR\) and \(TetR\)
- The binding affinities of \(NsrR\) and \( TetR \) to their respective binding sites.
How can such tuning be achieved?
- Expression rates of genes: There are 2 possible ways of tuning the expression rates of any particular gene.
- By inducing mutations in the promoter region of the gene that will alter its affinity towards RNA Polymerase in turn altering the rate of transcription and translation.
- By using the promoter regions of other known genes having transcription rates in the desired range. To get any desired expression rate, a combination of the above 2 methods could be used.
- Binding of \(TetR\) and \(NsrR\) to their respective binding sites (Operator region): This can be achieved using the followiing ways:
- Inducing mutations in the operator region so as to alter the affinities of the repressors to the binding sites.
- Inducing mutations in the repressor protiens by mutating the coding region that are responsible for producing these protiens. Hence, changing their affinities to the operator site.
Tuning the circuit:
Objective: To tune the circuit so that its activity is maximum between \(NO\) concentration of \(C_1\) and \(C_2\).
What parameters control the reactivity of the genetic circuit at different concentrations?
- Since \(TetR\) starts repressing the production of our Iron chelator Aerobactin after a certain concentration of \(NO\), it is evident that the parameters associated with the \(TetR\) system, control the higher bound of the circuit's active region (the concentration range of \(NO\) for which the expression of Aerobactin is maximum).
- Similarly, the NsrR represses the expression of Aerobatin at lower concentrations of \(NO\). Therefore, the parameters associated with the \(NsrR\) system will control the lower bound of the active region of our circuit.
We tuned the parameters of the \(TetR\) and the \(NsrR\) system so that the circuit is active between \(C_1\) and \(C_2\) concentrations of \(NO\). We got the following results:
| Parameter | Description | Value | Unit |
|---|---|---|---|
| \(\beta_{NsrR}\) | Constitutive rate of production of \(NsrR\) | \(7\) | \(\mu M \space min^{-1}\) |
| \(\beta_{NorR}\) | Constitutive rate of production of \(NorR\) | \(1\) | \(\mu M \space min^{-1}\) |
| \(\beta_{TetR}\) | Maximum rate of production of \(TetR\) | \(7.93\) | \(\mu M \space min^{-1}\) |
| \(K_{D_1}\) | Equilibrium constant of \(NsrR\) repression | \(0.01225\) | \(\mu M^2\) |
| \(K_{D_2}\) | Equilibrium constant of \(TetR\) repression | \(0.3136\) | \(\mu M^2\) |
How long does it take for our circuit to respond to the environment inside a Leishmania infected Macrophage?
Possible improvements to the Nitric Oxide Sensor
To create a better sensor, the sensitivity of the circuit should be very specific around the concentrations \(C_1\) and \(C_2\) of \(NO\). Ideally, the rate of Aerobactin production should change very steeply at the boundary regions of the active region of the circuit (\(NO\) concentration). The factor that controls the sensitivity of circuit is the number of molecules of the substrate binding to the other reactant in any reaction. This factor is equivalent to the Hill coefficient. We observe
References
- Aldridge, Christopher & Razzak, Anthony & Babcock, Tricia & Helton, william & Espat, N.. (2008). Lipopolysaccharide-Stimulated RAW 264.7 Macrophage Inducible Nitric Oxide Synthase and Nitric Oxide Production Is Decreased by an Omega-3 Fatty Acid Lipid Emulsion. The Journal of surgical research. 149. 296-302. 10.1016/j.jss.2007.12.758.