We will now look at the trigger from a diffusion perspective. This is different as in the earlier models, we focus on how cell populations change in time and derive health and collateral functions that show their dependence on the initial viral dose. We did not consider any spatial relationships between the cellular populations. Now we will do so and the discussion will give us another way to look at the nonlinear interactions between M and N. It is well known that second messenger systems often involve C
a
++ ion movement in and out of the cell. The amount of free C
a
++ ion in the cell is controlled by complicated mechanisms, but some is stored in buffer complexes. The release of calcium ion from these buffers plays a big role in cellular regulatory processes and which protein P(T
1) is actually created from a trigger T
0. The diffusion model is very powerful. Consider some substance u which satisfies
$$\begin{array}{@{}rcl@{}} \frac{\partial u }{ \partial t } &=& D \frac{\partial^{2} {u} }{ \partial {x}^{2} }\\ -D \frac{\partial u }{ \partial x } \mid_{0,L} &=& J_{0,L} \end{array} $$
where D is called the diffusion constant for this substance. For simplicity this is a one dimensional model where the spatial variable x comes from the line segment [0,L]. For example, instead of a three dimensional cell modeled as a sphere, the cell is modeled as a string of finite length. Hence, stuff can only enter the cell from either the right or the left. The condition \(-D \frac {\partial u }{ \partial x } \mid _{0,L} = J_{0,L}\) states that there are conditions on the flux of u through the boundary at x=−0 or x=L that must be satisfied. The term J
0,L
can be thought of as an injection of current into the cell. In this model, we think of the substance u as being in equilibrium in the cell and the current injection J
0,L
alters that equilibrium and the diffusion model, when solved, tells us what happens to u due to the sudden current injection at the boundary. The critical review on the control of free calcium in cellular processing in [6] notes the concentration of C
a
++ in the cell is controlled by the reversible binding of calcium ion to the buffer complexes, B
1,B
2 and so forth. There in general are quite a few different buffer complexes which all behave differently. These buffer molecules act as calcium ion sensors that, in a sense, decode the information contained in the calcium ion current injection and then pass on a decision to a target. Many of these targets can be proteins transcribed by accessing the genome. Hence, the P(T
1) could be a buffer molecule B
j
and so the trigger that causes the calcium current injection into the cell could influence the concentration of a buffer B
j
and therefore influence how it itself is decoded. In this situation, the boundary condition J
0,L
plays the role of the entry calcium current. Such a calcium ion input current through the membrane could be due to membrane depolarization causing an influx of calcium ions through the port or via ligand binding to a receptor which in turn indirectly increases free calcium ion in the cytosol. Such mechanisms involve the interplay between the way the endoplasmic reticulum handles release and uptake and also the storage buffers. This boundary current in general therefore determines the u(t,x) solution through the diffusion equation. The exact nature of this solution is determined by the receptor types, buffers and storage sites in the ER. Differences in the period and magnitude of the calcium current u(t,x) resulting from the input J
0,L
trigger different second messenger pathways. Hence, there are many possible outcomes due to a given input current J
0,L
.
We will now modify discussions in [7, 8] and [9] that show us how to model calcium ion movement in the cell to develop a model of trigger movement in and out of the cell populations M and N.
We assume the trigger enters the host and can be sequestered in some form in both M and N cell populations. We also assume the trigger has associated with some sort of diffusion process; letting u(t,x) be the concentration of the trigger in the host at time t and spatial position x, we posit u
t
=D
0
u
xx
. Also, note we present our arguments as if the host was one dimensional; i.e., the two cell populations lies along a one dimensional axis measured by the location variable x. This is, of course, very simplistic, but we only want to suggest some functional dependencies here, so it will suffice for our purposes. Let’s assume these two populations use or bind the trigger with binding rate constants \(k_{M}^{+}\) and \(k_{N}^{+}\) and disassociation rate constants \(k_{M}^{-} \) and \(k_{N}^{-}\). Let the total number of cells in the host be P. Then the fraction of cells in the M population is \(\frac {\boldsymbol {M}}{P}\) which we call C
M
and the fraction in the N population is \(\frac {\boldsymbol {M}}{P}\) which is denoted by C
N
. Hence, the number of cells that have not been exposed to the trigger is P−M(t,x)−N(t,x)=F(t,x).
Now, let u(t,x) be the concentration of free trigger in the host at (t,x). Some of the trigger has been used to create cells in the populations M and N, but the rest is unused. Hence, if C is a cell which has not been altered by the trigger, we have the reactions
$$\begin{array}{@{}rcl@{}} T + \: \boldsymbol{C} &\rightarrow_{k_{M}^{+}}& \boldsymbol{M}, \quad \boldsymbol{M} \rightarrow_{k_{M}^{-}} T \: + \: \boldsymbol{C}\\ T + \: \boldsymbol{C} &\rightarrow_{k_{N}^{+}}& \boldsymbol{N}, \quad \boldsymbol{N} \rightarrow_{k_{N}^{-}} T \: + \: \boldsymbol{C} \end{array} $$
where T is the trigger. Of course, the equations above depend on time and space, but we have not written in that dependence to avoid clutter. Note also, in this context, the backward reaction in which trigger is freed from the cellular populations M(t,x) and N(t,x) is typically that of lysis and so the backward rates are part of our nonlinear interaction model. We are just adding low level detail. The corresponding dynamics are
$$\begin{array}{@{}rcl@{}} \frac{d\left[T\right]}{dt} &=& - \: k_{M}^{+} \left[T\right]\left[\boldsymbol{C}\right] \: + \: k_{M}^{-} \left[\boldsymbol{M}\right],\\ \quad \frac{d\left[\boldsymbol{M}\right]}{dt} &=& \: k_{M}^{+} \left[T\right] \left[\boldsymbol{C}\right] \: - \: k_{M}^{-} \left[\boldsymbol{M}\right]\\ \frac{d\left[T\right]}{dt} &=& - \: k_{N}^{+} \left[T\right]\left[\boldsymbol{C}\right] \: + \: k_{N}^{-} \left[\boldsymbol{N}\right],\\ \quad \frac{d\left[\boldsymbol{N}\right]}{dt} &=& \: k_{N}^{+} \left[T\right] \left[\boldsymbol{N}\right] \: - \: k_{N}^{-} \left[\boldsymbol{N}\right] \end{array} $$
and we also know \(\left [\boldsymbol {C}\right ] = \frac {\boldsymbol {P} - \boldsymbol {M} - \boldsymbol {N}}{\boldsymbol {P}}\). The amount of trigger being freed from the M population is \(k_{M}^{-} \frac {\boldsymbol {M}}{\boldsymbol {P}} = k_{M}^{-} C_{M}\) and the amount being added to the M population is the amount of trigger not in the M state minus the amount of trigger already in the M state. This can be calculated at
$$\begin{array}{@{}rcl@{}} k_{M}^{+} \: u(t,x) \: \frac{\boldsymbol{P} - \boldsymbol{N}}{\boldsymbol{P}} \: - \: k_{M}^{+} \frac{\boldsymbol{M}}{\boldsymbol{P}} u(t,x) &=& k_{M}^{+} \left(\!\frac{\boldsymbol{P} - \boldsymbol{N}}{\boldsymbol{P}} - \frac{\boldsymbol{M}}{\boldsymbol{P}}\! \right) u(t,x) \end{array} $$
To make the manipulations easier, let \(B_{M} = 1 - \frac {\boldsymbol {N}}{\boldsymbol {P}}\) and \(B_{N} = 1 - \frac {\boldsymbol {M}}{\boldsymbol {P}}\). We can rewrite the equation above as
$$\begin{array}{@{}rcl@{}} k_{M}^{+} \left(\frac{\boldsymbol{P} - \boldsymbol{N}}{\boldsymbol{P}} - \frac{\boldsymbol{M}}{\boldsymbol{P}} \right) u(t,x) &=& k_{M}^{+} \left(B_{M} - C_{M} \right) u(t,x) \end{array} $$
A similar analysis gives the amount of trigger being added to the N population as
$$\begin{array}{@{}rcl@{}} k_{N}^{+} \left(\frac{\boldsymbol{P} - \boldsymbol{M}}{\boldsymbol{P}} - \frac{\boldsymbol{N}}{\boldsymbol{P}} \right) u(t,x) &=& k_{N}^{+} \left(B_{N} - C_{N} \right) u(t,x) \end{array} $$
Thus, the diffusion dynamics are
$$\begin{array}{@{}rcl@{}} \frac{\partial u}{\partial t} &=& k_{M}^{-} C_{M} - k_{M}^{+} \left(B_{M} - C_{M} \right) u(t,x)\\&& +\: k_{N}^{-} C_{N} - k_{N}^{+} \left(B_{N} - C_{N} \right) u(t,x) + D_{0} \frac{\partial^{2} u}{\partial x^{2}} \end{array} $$
where D
0 is diffusion coefficient for free trigger. We now assume the spread of cells we collect into the populations M and N satisfy some sort of diffusion law. Certainly, cells are added to these populations as the trigger diffuses throughout the host’s body. Hence, this assumption is a good start. We therefore assume the diffusion dynamics for C
M
and C
N
are given by
$$\begin{array}{@{}rcl@{}} \frac{\partial C_{M}}{\partial t} &=& - k_{M}^{-} C_{M} \: + \: k_{M}^{+} \left(B_{M} - C_{M} \right) u(t,x) + D_{M} \frac{\partial^{2} C_{M}}{\partial x^{2}}\\ \frac{\partial C_{N}}{\partial t} &=& - k_{N}^{-} C_{N} \: + \: k_{N}^{+} \left(B_{N} - C_{N} \right) u(t,x) + D_{N} \frac{\partial^{2} C_{N}}{\partial x^{2}} \end{array} $$
Now consider the free trigger plus a correction due to the population fractions C
M
and C
N
. We denote this by w(t,x) and note that w=u + C
M
+C
N
and
$$\begin{array}{@{}rcl@{}} w_{xx} &=& u_{xx} + (C_{M})_{xx} + (C_{N})_{xx} \: \Longrightarrow u_{xx}\\ &=& w_{xx} - (C_{M})_{xx} - (C_{N})_{xx}. \end{array} $$
Thus,
$$\begin{array}{@{}rcl@{}} \frac{\partial w}{\partial t} &=& u_{t} + (C_{M})_{t} + (C_{N})_{t} = k_{M}^{-} C_{M}\\&& - k_{M}^{+} (B_{M} - C_{M}) (w - C_{M} - C_{N})\\&& + k_{N}^{-} C_{N} - k_{N}^{+} (B_{N} - C_{N}) (w - C_{M} - C_{N}) + D_{0} \: u_{xx}\\ & & -k_{M}^{-} C_{M} + k_{M}^{+} (B_{M} - C_{M}) (w - C_{M} - C_{N}) + D_{M} (C_{M})_{xx} \\ & & - k_{N}^{-} C_{N} + k_{N}^{+} (B_{N} - C_{N}) (w - C_{M} - C_{N}) + D_{N} (C_{M})_{xx} \end{array} $$
This simplifies to
$$\begin{array}{@{}rcl@{}} \frac{\partial w}{\partial t} &=& D_{0} \:u_{xx} + D_{M} (C_{M})_{xx} + D_{N} (C_{N})_{xx}\\ &=& D_{0} \: (w_{xx} - (C_{M})_{xx} - (C_{N})_{xx}) + D_{M} (C_{M})_{xx}\\ &&+\: D_{N} (C_{N})_{xx}\\ &=& D_{0} \: w_{xx} + (D_{0} - D_{M}) (C_{M})_{xx} + (D_{0} - D_{N}) (C_{N})_{xx} \end{array} $$
Thus, w satisfies
$$\begin{array}{@{}rcl@{}} w_{t} &=& D_{0} w_{xx} + (D_{M} - D_{0}) (C_{M})_{xx}\\ &&+ (D_{N} - D_{0}) (C_{N})_{xx}, -D_{0} w_{x}\left|{\!~\!}_{0,L} = J_{0,L}\right. \end{array} $$
(4)
where we have not shown the derivation of the boundary terms here as they are less germane to our interests. It seems reasonable to assume that the interaction with the cell populations, determined by \(k_{M}^{-}\) and \(k_{M}^{+} u\) is fast and reaches equilibrium quickly. Hence, we will assume that (C
M
)
t
=(C
M
)
xx
=0 and (C
N
)
t
=(C
N
)
xx
=0 giving
$$\begin{array}{@{}rcl@{}} -k_{M}^{-} C_{M} = k_{M}^{+} \left(B_{M} - C_{M} \right) u, \quad -k_{N}^{-} C_{N} = k_{N}^{+} \left(B_{N} - C_{N} \right) u \end{array} $$
Solving, we find
$$\begin{array}{@{}rcl@{}} C_{M} &=& \frac{k_{M}^{+} B_{M} u}{k_{M}^{-} + k_{M}^{+} u}\\ C_{N} &=& \frac{k_{N}^{+} B_{N} u}{k_{N}^{-} + k_{N}^{+} u} \end{array} $$
Now define \(K_{M} = \frac {}{}\) and \(K_{N} = \frac {}{}\). We can then rewrite our equations as
$$\begin{array}{@{}rcl@{}} C_{M} &=& \frac{B_{M} u}{K_{M} + u} \Longrightarrow C_{M} (K_{M} + u) = B_{M} u \end{array} $$
(5)
$$\begin{array}{@{}rcl@{}} C_{N} &=& \frac{B_{N} u}{K_{N} + u} \Longrightarrow C_{N} (K_{N} + u) = B_{N} u \end{array} $$
(6)
Plugging in for u, we have
$$\begin{array}{@{}rcl@{}} C_{M} K_{M} &+& (C_{M} - B_{M}) (w - C_{M} - C_{N}),\\ \quad C_{N} K_{N}&+& (C_{N} - B_{N}) (w - C_{M} - C_{N}) \end{array} $$
Then, another rewrite gives
$$\begin{array}{@{}rcl@{}} B_{M} w &=& C_{M} (K_{M} + B_{M} + w) - C_{M}^{2} + B_{M} C_{N}\\ B_{N} w &=& C_{N} (K_{N} + B_{N} + w) - C_{N}^{2} + B_{N} C_{M} \end{array} $$
Note this tells us that C
M
and C
N
are functions of the w Let C
M
(w) and C
N
(w)denote this functional dependence. From the chain rule, we have \(\frac {\partial C_{M} }{ \partial x } = \frac {\partial C_{M} }{ \partial w } \: \frac {\partial w }{ \partial x }\) and \(\frac {\partial C_{N} }{ \partial x } = \frac {\partial C_{N} }{ \partial w } \: \frac {\partial w }{ \partial x }\). Then, the dynamics become
$$\begin{array}{@{}rcl@{}} w_{t} &=& \frac{\partial^{2} {w} }{ \partial {x}^{2} } + \frac{\partial}{\partial x} \left(\! (D_{M} - D_{0}) \frac{\partial C_{M} }{ \partial w } \frac{\partial w }{ \partial x } + (D_{N} - D_{0}) \frac{\partial C_{N} }{ \partial w } \frac{\partial w }{ \partial x } \right)\\ &=& \frac{\partial}{\partial x} \left(\left(D_{0} + (D_{M} - D_{0}) \frac{\partial C_{M} }{ \partial w } + (D_{N} - D_{0}) \frac{\partial C_{N} }{ \partial w } \right) \frac{\partial w }{ \partial x } \right) \end{array} $$
Notice that if we define a new diffusion coefficient, 
for the diffusion process that governs w by
$$\begin{array}{@{}rcl@{}} \mathcal{D} &=& D_{0} + (D_{M}- D_{0}) \frac{\partial C_{M} }{ \partial w } ++ (D_{N}- D_{0}) \frac{\partial C_{N} }{ \partial w } \end{array} $$
(7)
we obtain
Further approximations
What would it mean if u≪K
M
and u≪K
N
? Let’s take the first case: u≪K
M
implies \(k_{M}^{+} u \ll k_{M}^{-}\). Now \(k_{M}^{-}\) represents the rate that the M cells lose the trigger. This occurs only when the cells are destroyed. These cells are destroyed either because they have exceeded their normal lifespans or because the trigger has made them more fragile. This fragility could mean the cells have caught the attention of the immune system and are being destroyed or the fragility of the cells is such that standard apoptosis strategies are employed to remove the damaged cell. Note losing trigger from the M cells therefore corresponds to increasing trigger concentration. Since \(k_{M}^{+}\) is the rate at which M cells are formed, \(k_{M}^{+} u\) giving the amount of trigger lost from the formation of the M cells per M cell. We generally assume that in a trigger situation, the trigger is growing inside the M and N cell populations. If the trigger is a virus, replication only occurs after infection and once inside these cells, the virus grows. This additional trigger is then released into the host upon lysis of a M or N cell. It seems reasonable to assume the amount of released trigger is usually quite a bit bigger than the amount of trigger that initiates the formation of these cells. Thus, the inequality \(k_{M}^{+} u \ll k_{M}^{-}\) seems reasonable. A similar argument shows that \(k_{N}^{+} u \ll k_{N}^{-}\). Thus, it is reasonable to assume u≪K
m
and u≪K
N
which leads to the approximations
$$\begin{array}{@{}rcl@{}} C_{M} &=& \frac{B_{M}}{K_{M}} \: u, \quad C_{N} = \frac{B_{N}}{K_{N}} \: u \end{array} $$
(8)
Thus, w has become
$$\begin{array}{@{}rcl@{}} w &=& \left(1 + \frac{B_{M}}{K_{M}} + \frac{B_{N}}{K_{N}} \right) \: u \Longrightarrow u = \frac{w}{1 + \frac{B_{M}}{K_{M}} + \frac{B_{N}}{K_{N}}} \end{array} $$
We can then rewrite C
M
and C
N
as
$$\begin{array}{@{}rcl@{}} C_{M} &=& \frac{B_{M}}{K_{M}} \: \frac{w}{1 + \frac{B_{M}}{K_{M}} + \frac{B_{N}}{K_{N}}}, \quad \quad C_{N} = \frac{B_{N}}{K_{N}} \: \frac{w}{1 + \frac{B_{M}}{K_{M}} + \frac{B_{N}}{K_{N}}} \end{array} $$
or
$$\begin{array}{@{}rcl@{}} C_{M} &=& \frac{\frac{B_{M}}{K_{M}} } { 1 + \frac{B_{M}}{K_{M}} + \frac{B_{N}}{K_{N}}} \: w, \quad \quad C_{N} = \frac{\frac{B_{N}}{K_{N}}} { 1 + \frac{B_{M}}{K_{M}} + \frac{B_{N}}{K_{N}}} \: w \end{array} $$
We can find the partials with respect to w:
$$\begin{array}{@{}rcl@{}} (C_{M})_{w} &=& \frac{\frac{B_{M}}{K_{M}}}{1 + \frac{B_{M}}{K_{M}} + \frac{B_{N}}{K_{N}}}, \quad \quad (C_{N})_{w} = \frac{\frac{B_{N}}{K_{N}}}{ 1 + \frac{B_{M}}{K_{M}} + \frac{B_{N}}{K_{N}} } \end{array} $$
Then, letting \(\gamma _{M}= \frac {B_{M}}{K_{M}}\) and \(\gamma _{N} = \frac {B_{N}}{K_{N}}\), we have
$$\begin{array}{@{}rcl@{}} (C_{M})_{w} &=& \frac{\gamma_{M}}{1 + \gamma_{M} + \gamma_{N}}, \quad \quad (C_{N})_{w} = \frac{\gamma_{N}}{ 1 + \gamma_{M} + \gamma_{N}} \end{array} $$
We can then redo our calculations for w
t
.
$$\begin{array}{@{}rcl@{}} w_{t} &=& \frac{\partial}{\partial x} \left\{ \left(D_{0} + (D_{M} - D_{0}) \left(\frac{\gamma_{M}}{1 + \gamma_{M} + \gamma_{N}}\right)\right.\right.\\ &&+ \left. \left.(D_{N} - D_{0}) \left(\frac{\gamma_{N}}{ 1 + \gamma_{M} + \gamma_{N}}\right) \right) \frac{\partial w }{ \partial x } \right\} \end{array} $$
Letting Λ denote the term 1+γ
M
+γ
N
, then w=Λ
u and so we have
$$\begin{array}{@{}rcl@{}} w_{t} &=& \Lambda u_{t}, \quad \quad w_{x} = \Lambda u_{x}, \quad \quad w_{xx} = \Lambda u_{xx}. \end{array} $$
We also have
$$\begin{array}{@{}rcl@{}} w_{t} &=& \frac{D_{0} + D_{M} \gamma_{M} + D_{N} \gamma_{N}}{\Lambda} \: w_{xx}\\ &=& (D_{0} + D_{M} \gamma_{M} + D_{N} \gamma_{N}) u_{xx} \end{array} $$
Thus,
$$\begin{array}{@{}rcl@{}} \Lambda u_{t} &=& (D_{0} + D_{M} \gamma_{M} + D_{N} \gamma_{N}) \: u_{xx} \Longrightarrow u_{t}\\ &=& \frac{\Lambda \: D_{0} + D_{M} \gamma_{M} + D_{N} \gamma_{N}} {\Lambda} \: u_{xx} \end{array} $$
Define the new diffusion constant \(\hat {\mathcal {D}}\) by \(\hat {\mathcal {D}} = \frac {D_{0} + D_{M} \gamma _{M} + D_{N} \gamma _{N}} {\Lambda }\). The free trigger dynamics are thus 
.
Approximations to trigger modification
Let’s examine what might happen if a trigger event T
0 initiated an increase in M. This trigger event initiates a complex pathway culminating in a protein transcription (see Section “General trigger models” for the general trigger discussion). Recall, we let the pathway leading to a change in fragility for M be \(\boldsymbol {\mathcal {P_{M}}}\). We associate such a change in fragility with the alteration of a subsidiary signal T
M
and we derived
$$\begin{array}{@{}rcl@{}} \delta_{T_{M}} &=& \mu_{M} \: \left(2 \epsilon_{M}\: + \: \epsilon_{M}^{2}\right) \end{array} $$
for parameters μ
M
and ε
M
. In Section “Computational abstractions”, a computational approach using sigmoid activation function h
M
further found if the change in fragility in M due to the signal T
M
is the alteration of a protein or protein complex called P(T
M
), then
$$\begin{array}{@{}rcl@{}} h_{\boldsymbol{M}}(\left[P(T_{M})\right]) &=& \frac{e_{P(T_{M})}}{2} \: \delta_{P(T_{M})} \: \boldsymbol{P(T_{M})}^{max}\\ && \left(1 + \tanh\left(\frac{\left[P(T_{M})\right]}{g_{\boldsymbol{P(T_{M})}}}\right) \right) \end{array} $$
where \(e_{P(T_{M})}\) is a scaling factor and \(\delta _{P(T_{M})}\) is the change in P(T
M
) expression. Our model of the change in maximum P(T
M
) expression is therefore \(\Delta \: \boldsymbol {P(T_{M})}^{max} \: = \: h_{\boldsymbol {P(T_{M})}}([P(T_{M})])\).
Let’s assume C
M
is increased to C
M
+ε. It is reasonable to assume that both \(k_{M}^{+}\) and \(k_{M}^{-}\) are independent of the amount of C
M
that is present. The same comment holds for \(k_{N}^{+}\) and \(k_{N}^{-}\). We have \(B_{M} = 1 - \frac {N}{P}\) stays the same, but \(B_{N} = 1 - \frac {M}{P} = 1 - C_{M} \rightarrow 1 - C_{M} - \epsilon = B_{N} - \epsilon \). Thus,
$$\begin{array}{@{}rcl@{}} \Lambda &=& 1 + \frac{B_{M}}{K_{M}} + \frac{B_{N}}{K_{M}} \rightarrow 1 + \frac{B_{M}}{K_{M}} + \frac{B_{N}}{K_{N}} - \frac{\epsilon}{K_{N}}. \end{array} $$
Thus, the new value is \(\hat {\Lambda } = \Lambda - \frac {\epsilon }{K_{N}}\). This implies
$$\begin{array}{@{}rcl@{}} \hat{\mathcal{D}} &=& \frac{D_{0} + D_{M} \frac{B_{M}}{K_{M}} + D_{N} \frac{B_{N}}{K_{N}} }{\Lambda} \rightarrow \frac{D_{0} + D_{M} \frac{B_{M}}{K_{M}} + D_{N} \frac{B_{N}}{K_{N}} - D_{N} \frac{\epsilon}{K_{N}}}{\Lambda- \frac{\epsilon}{K_{N}}} \end{array} $$
\(\tilde {\mathcal {D}} = \frac {\Lambda \: \hat {\mathcal {D}} - \epsilon \frac {D_{N}}{K_{N}}}{\Lambda - \epsilon \frac {1}{K_{N}}}\). Now let ξ
M
denote \(\frac {1}{K_{N}}\) and use that in the equation above. We find \(\tilde {\mathcal {D}} = \frac {\Lambda \: \hat {\mathcal {D}} - \epsilon \xi _{M} D_{N} }{\Lambda - \epsilon \xi _{M} }\). The change in the diffusion constant is then
$$\begin{array}{@{}rcl@{}} \Delta \hat{\mathcal{D}} &=&\hat{\mathcal{D}} - \frac{\Lambda \: \hat{\mathcal{D}} - \epsilon \xi_{M} D_{N} }{\Lambda - \epsilon \xi_{M} } = \frac{\epsilon D_{N}}{\Lambda - \epsilon \xi_{M}} \: \left(1 - \frac{\hat{\mathcal{D}}}{D_{N}} \right) \end{array} $$
To first order, we know \(\frac {\epsilon }{\Lambda - \epsilon \xi _{M}} \approx \frac {\epsilon }{\Lambda }\) and so \(\Delta \hat {\mathcal {D}} \approx \frac {\epsilon D_{N}}{\Lambda }\left (1 - \frac {\hat {\mathcal {D}}}{D_{N}} \right)\). We conclude the new diffusion dynamics are on the order of
$$\begin{array}{@{}rcl@{}} u_{t} &=& \left(\hat{\mathcal{D}} + \Delta \hat{\mathcal{D}} \right) u_{xx} \approx \left(\hat{\mathcal{D}} + \epsilon \frac{D_{N}}{\Lambda} \left(1 - \frac{\hat{\mathcal{D}}}{D_{N}} \right) u_{xx} \right. \end{array} $$
This change in the solution u(t,x) then can then initiate further changes in the distribution of the M and N cells. A similar argument can be used for a change in the N population and it is clear if the signal generates changes in both M and N, to first order we generate an altered diffusion model whose solution gives us clues as to new behavior. Without boundary conditions, the general solution to a diffusion model with diffusion constant D is given by
$$\begin{array}{@{}rcl@{}} \phi(t,x) &=& \frac{1}{\sqrt{4 \pi D \: t}} \: e^{-\frac{x^{2}}{4Dt}} \end{array} $$
Hence, our usual trigger solution is \(u(t,x) = \frac {1}{\sqrt {4 \pi \hat {\mathcal {D}} \: t}} \: e^{-\frac {x^{2}}{4 \hat {\mathcal {D}} t}}\) which is altered to \(\hat {u}(t,x) = \frac {1}{\sqrt {4 \pi \tilde {\mathcal {D}} \: t}} \: e^{-\frac {x^{2}}{4 \tilde {\mathcal {D}} t}}\).
to form the complex 
. This chain of events looks like this: 
where CSR denotes a cell surface receptor. 
then acts to phosphorylate another protein. The cell is filled with large amounts of a transcription factor we will denote by T
1 and an inhibitory protein for T
1 we label as 
.
for the diffusion process that governs w by
.
