User:Johnsy/SB Mini iGEM

The Model
In this section, we present a mathematical model of the system we are trying to develop. To simplify the system, we split the system into three components:


 * 1) The Cell Free System
 * 2) The Xenopus Oocyte ACh Receptors
 * 3) Voltage-Gated Ion Channel Response

The Cell Free System
We have decided to use microfludic devices and use a cell free system because parameters in such a system are more controllable than in a normal E. coli system. To design a good model, we must first understand the biochemistry behind the system which is outlined in previous sections and summarized here.

Our input of AHL activates a LuxR promoter which then allows transcription of the enzyme PEMT, the rate limiting step of the synthesis of ACh. Although there are several more enzymes invovled in this pathway and they are all under the control of the LuxR promoter, we have decided only consider the transcription and expression of PEMT and assume that the production of all other enzyems are on the same order of time and the rate of production of ACh is only dependent upon the activity of PEMT.

The enzyme PEMT itself can be modelled with Michaelis-Menten kinetics whose parameters are well publicized. With these biochemical systems in mind, we can come up with the following system of differential equations to model our system.

$$ \begin{alignat}{2} \frac{d[mRNA]}{dt} & = \frac{k_1[AHL]^n}{K_{m1}^n+[AHL]^n} - d_1[mRNA] \\ \frac{d[PEMT]}{dt} & = k_2[mRNA] - d_2[PEMT] \\ \frac{d[ACh]}{dt} & = \frac{k_3[PEMT][PMME]}{K_{m2}+[PMME]} - k_4[ACh] \\ \end{alignat} $$

Looking at the first equation that governs the expression of mRNA, the source term is a hill activation equation with AHL activating the transcription of the PEMT gene. The sink term represents the degradation of the mRNA. Since the degradation of the mRNA is slower than the flushing rate, we assume that this term is primarily affected by the rate of flushing that we induce in our microfludic system. In the second equation, the rate of production of PEMT enzyme is related to the amount of mRNA in the system as well as the degradation of the enzyme. We must remember that the degradation term of the enzyme can be altered by changing the rate that we flush our system out. In the thrid equation, the concentration of ACh is only governed by a Michaelis-Menten equation depending on the concentration of enzyme. We put the concentration level in, even though it is kept constant in the normal Michaelis-Menten equation since over time, we are changing the level of enzyme in the system. We must also put a degradation term for the ACh, and like the enzyme and mRNA, is dependent upon the rate of flushing that we use for our microfludic device.

The parameters for the above system are explained below.
 * $$k_1$$ - the transcription rate
 * $$K_{m1}$$ - DNA binding constant
 * $$n$$ - Hill cooperativity factor for activator
 * $$d_1$$ - degradation rate/flushing rate of mRNA
 * $$k_2$$ - the translation rate
 * $$d_2$$ - degradation rate/flushing rate of PEMT
 * $$k_3$$ - the rate constant for the enzymatic reaction of PEMT
 * $$K_{m2}$$ - Michaeli-Menten constant (Km) for PEMT
 * $$k_4$$ - degradation rate/flushing rate of ACh

We are also assuming that the rate of PMME, the precursor of ACh is kept constant and at a high level enough so that the limiting factor for the production of ACh is the enzyme concentration.

To come up with a transfer function of the system, we can look at the system at steady state and assume the quasi-steady state approximation for some of the variables. First, we assume that the level of mRNA in the system is kept constant at steady state. This means that $$\frac{d[mRNA]}{dt} = 0$$. Rearranging the equation, we get the following equation.

$$[mRNA]^* = \frac{k_1}{d_1} \frac{[AHL]^n}{K_{m1}^n+[AHL]^n}$$

We can substitute this equation into the differential equation for PEMT to get the following equation.

$$\frac{d[PEMT]}{dt} = \frac{k_1k_2}{d_1} \frac{[AHL]^n}{K_{m1}^n+[AHL]^n} - d_2[PEMT]$$

Again, at if we wait long enough, we can assume that the concentration of enzyme is also constant at long periods of time. As with the mRNA assumption, we can find the fixed point of the system to be the following:

$$[PEMT]^* = \frac{k_1k_2}{d_1d_2} \frac{[AHL]^n}{K_{m1}^n+[AHL]^n}$$

And finally, we can substitute this expression back into the differential equation for ACh. We can then solve for the fixed point of ACh to get the transfer function between AHL levels and AChl levels.

$$\frac{d[ACh]}{dt} = \frac{k_1k_2k_3}{d_1d_2}\frac{[AHL]^n}{K_{m1}^n+[AHL]^n}\frac{[PMME]}{K_{m2}+[PMME]} - k_4[ACh]$$

$$[ACh]^* = \frac{k_1k_2k_3}{k_4d_1d_2}\frac{[AHL]^n}{K_{m1}^n+[AHL]^n}\frac{[PMME]}{K_{m2}+[PMME]}$$

Now that we have a relationship between AHL levels that we induce our system with the the ACh levels as our output, we can now take this and see how this affects the injected current that we use to induce the voltage gated ion channels.

Xenopus Oocyte ACh Receptors
ACh recptors have been expressed in Xenopus oocyte cells and have been well characterized. From Shields, et al, we have obtained the relationship between the agonist for the receptor (here it is ACh) and the observed current that is generated. We have to remember that the nicotinic recpetor is itself a ACh dependent sodium channel which generates a current by allowing sodium ions to exit the cell. We assume that this current is "injected" or used to stimulate the other voltage-gated ion channels that are expressed on the surface of the Xenopus oocyte cell (both sodium and potassium - discussed later).

The simple relationship reflects the cooperativity of the binding necessary to activate the receptor and is represented as a Hill equation seen below.

$$I_{nAChR} = N\frac{I_{max}[ACh]^n}{[ACh]^n + EC_{50}^n}$$

The parameters are defined below:
 * $$N$$ - the number of receptors on the surface of the cell
 * $$I_{max}$$ - the maximum current that is generated by each receptor
 * $$n$$ - Hill cooperativity factor for the receptor
 * $$EC_{50}$$ - the concentration of the agonist which causes half maximal response

Coupling this equation with the above transfer function between AHL and ACh, we can establish the relationship between AHL concentrations and the injected current produced by the nicotinic receptors.

Voltage Gated Ion Channel Response
The voltage gated ion channels can be modelled using a Hodgkin-Huxley like model since we are also using potassium and sodium ion channels that we will express on the surface of the oocyte. We will measure the potential difference using patch clamps or similar apparatus. We decided to use the Hodgkin-Huxley equations since they are well characterized and are known to accurately model the behavior of neurons.

The Wilson model, a simplified Hodgkin-Huxley model based on neocortical neurons is shown below. Two equations relating T and H are added in, reflecting an approximation to Ca2+ current and Ca2+ mediated K+ hyperpolarization current respectively.

$$ \begin{alignat}{2} \frac{dV}{dt} & = -m_\infty [V](V-0.5)-26R(V+0.95)-g_TT(V-1.2)-g_HH(V+0.95)+I_{nAChR} \\ \frac{dR}{dt} & = \frac{1}{\tau_R}(-R + R_\infty [V]) \\ \frac{dT}{dt} & = \frac{1}{14}(-T + T_\infty [V]) \\ \frac{dH}{dt} & = \frac{1}{45}(-H+3T) \\ \end{alignat} $$

The values for $$m_\infty[V]$$, $$R_\infty[V]$$, and $$T_\infty[V]$$ are shown below and have been extracted from experimental values reported in Wilson's paper.

$$ \begin{alignat}{2} m_\infty[V] & = 17.8+47.6V+33.8V^2 \\ R_\infty[V] & = 1.24+3.7V+3.2V^2 \\ T_\infty[V] & = 8(V+0.725)^2 \\ \end{alignat} $$

The parameters of the equations are defined below:


 * $$V$$ - the time dependent membrane potential
 * $$m$$ - Na+ channel activation
 * $$h$$ - Na+ channel inactivation
 * $$n$$ - K+ channel activatioin
 * $$E_{Na}$$ - Equilibrium potential for Na+
 * $$E_{K}$$ - Equilibrium potential for K+
 * $$m_\infty[V]$$ - Na+ activation function
 * $$\tau_R$$ - time constant for K+ channel activation
 * $$R_\infty[V]$$ - Equilibrium state of K+ channel activation
 * $$g_k$$ - conductance from K+ ions

With matlab, we can see the firing rate when we alter the parameters of the equation and sample this firing rate getting a transfer function between injected current to firing rate. With this, we can easily couple the other parts of our system to get a complete transfer function between the concentration of AHL to firing rate which we can measure electrically to have a varying output signal.