Mixing
Analyzing behavior around a maximum without explicitly solving a differential equation
$begingroup$
A pretty standard biological reaction is that of an enzyme on it's target which might have the following differential equation to describe it's evolution:
begin{align}
dot T &= D - gamma T \
dot M &= D - gamma M - frac{kTM}{K + M }
end{align}
In this reaction species $T$ is being produced and enzymatically degrades $M$. This kind of equation produces "pulse" behavoir, and there is a distinct maximum in $M$ that I would like to analyze - what does it depend on?
Unfortunately this kind off differential equation seems pretty hard to solve. We can solve for $T = D/gamma (1 - e^{-gamma t}) $, but the $M$ equation is difficult. If you solve for $dot M = 0$, you get
$$ gamma M^2 + ( gamma K_M - D + k frac{D}{gamma}(1- e^{-gamma t})) M - D K = 0 $$
This is an equation that relates $M$ to $t$ at the max, but I need another constraint to solve for $M$ and $t$. Normally I guess this would come from solving the differential equation, but I'm not sure a closed form exists in this case. I'm just trying to get a closed form for the maximum $M$, is this possible without solving the equation?
Edit: I guess you shouldn't be able to do this without taking into account the initial conditions, maybe this isn't possible.
ordinary-differential-equations nonlinear-system
asked Jan 24 at 23:13
Mike FlynnMike Flynn
6321617
$endgroup$
add a comment |
$begingroup$
A pretty standard biological reaction is that of an enzyme on it's target which might have the following differential equation to describe it's evolution:
begin{align}
dot T &= D - gamma T \
dot M &= D - gamma M - frac{kTM}{K + M }
end{align}
In this reaction species $T$ is being produced and enzymatically degrades $M$. This kind of equation produces "pulse" behavoir, and there is a distinct maximum in $M$ that I would like to analyze - what does it depend on?
Unfortunately this kind off differential equation seems pretty hard to solve. We can solve for $T = D/gamma (1 - e^{-gamma t}) $, but the $M$ equation is difficult. If you solve for $dot M = 0$, you get
$$ gamma M^2 + ( gamma K_M - D + k frac{D}{gamma}(1- e^{-gamma t})) M - D K = 0 $$
This is an equation that relates $M$ to $t$ at the max, but I need another constraint to solve for $M$ and $t$. Normally I guess this would come from solving the differential equation, but I'm not sure a closed form exists in this case. I'm just trying to get a closed form for the maximum $M$, is this possible without solving the equation?
Edit: I guess you shouldn't be able to do this without taking into account the initial conditions, maybe this isn't possible.
ordinary-differential-equations nonlinear-system
asked Jan 24 at 23:13
Mike FlynnMike Flynn
6321617
$endgroup$
$begingroup$
So what kind of analysis of the maximum do you want? The value itself? Or the time at which it happens? Surely, it depends on the initial conditions since different trajectories of ODE intersect nullcline $dot{M} = 0$ at different points. You can get just any of them by starting close to it.
$endgroup$
– Evgeny
Jan 27 at 18:27
$begingroup$
You are correct. I tried solving this problem by linking the initial conditions to the maximum via taylor series, but that results in an intractable polynomial-exponential mix (I also had to expand to 6 terms). I just want a simple expression for the max :(
$endgroup$
– Mike Flynn
Jan 28 at 19:17
$begingroup$
I doubt that it is possible to find such expression, but I might be wrong... At least I don't see any good idea for how this could be find without numerical tabulation or solving the ODE. I see a sort of workaround for your question if you consider $T(0)$ and "time $t^ast$ to reach maximum $M$" as your initial data. In that case you solve for $M$ in equation $dot{M} = 0$ substituting $T$ with solution to $dot{T} = D - gamma T$ at $t = t^ast$ with known $T(0)$. But I'm not sure that this is what you need.
$endgroup$
– Evgeny
Jan 28 at 20:55
add a comment |
$begingroup$
A pretty standard biological reaction is that of an enzyme on it's target which might have the following differential equation to describe it's evolution:
begin{align}
dot T &= D - gamma T \
dot M &= D - gamma M - frac{kTM}{K + M }
end{align}
In this reaction species $T$ is being produced and enzymatically degrades $M$. This kind of equation produces "pulse" behavoir, and there is a distinct maximum in $M$ that I would like to analyze - what does it depend on?
Unfortunately this kind off differential equation seems pretty hard to solve. We can solve for $T = D/gamma (1 - e^{-gamma t}) $, but the $M$ equation is difficult. If you solve for $dot M = 0$, you get
$$ gamma M^2 + ( gamma K_M - D + k frac{D}{gamma}(1- e^{-gamma t})) M - D K = 0 $$
This is an equation that relates $M$ to $t$ at the max, but I need another constraint to solve for $M$ and $t$. Normally I guess this would come from solving the differential equation, but I'm not sure a closed form exists in this case. I'm just trying to get a closed form for the maximum $M$, is this possible without solving the equation?
Edit: I guess you shouldn't be able to do this without taking into account the initial conditions, maybe this isn't possible.
ordinary-differential-equations nonlinear-system
asked Jan 24 at 23:13
Mike FlynnMike Flynn
6321617
$endgroup$
A pretty standard biological reaction is that of an enzyme on it's target which might have the following differential equation to describe it's evolution:
begin{align}
dot T &= D - gamma T \
dot M &= D - gamma M - frac{kTM}{K + M }
end{align}
In this reaction species $T$ is being produced and enzymatically degrades $M$. This kind of equation produces "pulse" behavoir, and there is a distinct maximum in $M$ that I would like to analyze - what does it depend on?
Unfortunately this kind off differential equation seems pretty hard to solve. We can solve for $T = D/gamma (1 - e^{-gamma t}) $, but the $M$ equation is difficult. If you solve for $dot M = 0$, you get
$$ gamma M^2 + ( gamma K_M - D + k frac{D}{gamma}(1- e^{-gamma t})) M - D K = 0 $$
This is an equation that relates $M$ to $t$ at the max, but I need another constraint to solve for $M$ and $t$. Normally I guess this would come from solving the differential equation, but I'm not sure a closed form exists in this case. I'm just trying to get a closed form for the maximum $M$, is this possible without solving the equation?
Edit: I guess you shouldn't be able to do this without taking into account the initial conditions, maybe this isn't possible.
ordinary-differential-equations nonlinear-system
ordinary-differential-equations nonlinear-system
asked Jan 24 at 23:13
Mike FlynnMike Flynn
6321617
asked Jan 24 at 23:13
Mike FlynnMike Flynn
6321617
edited Jan 24 at 23:58
Mike Flynn
asked Jan 24 at 23:13
Mike FlynnMike Flynn
6321617
asked Jan 24 at 23:13
Mike FlynnMike Flynn
6321617
asked Jan 24 at 23:13
Mike FlynnMike Flynn
6321617
6321617
$begingroup$
So what kind of analysis of the maximum do you want? The value itself? Or the time at which it happens? Surely, it depends on the initial conditions since different trajectories of ODE intersect nullcline $dot{M} = 0$ at different points. You can get just any of them by starting close to it.
$endgroup$
– Evgeny
Jan 27 at 18:27
$begingroup$
You are correct. I tried solving this problem by linking the initial conditions to the maximum via taylor series, but that results in an intractable polynomial-exponential mix (I also had to expand to 6 terms). I just want a simple expression for the max :(
$endgroup$
– Mike Flynn
Jan 28 at 19:17
$begingroup$
I doubt that it is possible to find such expression, but I might be wrong... At least I don't see any good idea for how this could be find without numerical tabulation or solving the ODE. I see a sort of workaround for your question if you consider $T(0)$ and "time $t^ast$ to reach maximum $M$" as your initial data. In that case you solve for $M$ in equation $dot{M} = 0$ substituting $T$ with solution to $dot{T} = D - gamma T$ at $t = t^ast$ with known $T(0)$. But I'm not sure that this is what you need.
$endgroup$
– Evgeny
Jan 28 at 20:55
add a comment |
$begingroup$
So what kind of analysis of the maximum do you want? The value itself? Or the time at which it happens? Surely, it depends on the initial conditions since different trajectories of ODE intersect nullcline $dot{M} = 0$ at different points. You can get just any of them by starting close to it.
$endgroup$
– Evgeny
Jan 27 at 18:27
$begingroup$
You are correct. I tried solving this problem by linking the initial conditions to the maximum via taylor series, but that results in an intractable polynomial-exponential mix (I also had to expand to 6 terms). I just want a simple expression for the max :(
$endgroup$
– Mike Flynn
Jan 28 at 19:17
$begingroup$
I doubt that it is possible to find such expression, but I might be wrong... At least I don't see any good idea for how this could be find without numerical tabulation or solving the ODE. I see a sort of workaround for your question if you consider $T(0)$ and "time $t^ast$ to reach maximum $M$" as your initial data. In that case you solve for $M$ in equation $dot{M} = 0$ substituting $T$ with solution to $dot{T} = D - gamma T$ at $t = t^ast$ with known $T(0)$. But I'm not sure that this is what you need.
$endgroup$
– Evgeny
Jan 28 at 20:55
$begingroup$
So what kind of analysis of the maximum do you want? The value itself? Or the time at which it happens? Surely, it depends on the initial conditions since different trajectories of ODE intersect nullcline $dot{M} = 0$ at different points. You can get just any of them by starting close to it.
$endgroup$
– Evgeny
Jan 27 at 18:27
$begingroup$
So what kind of analysis of the maximum do you want? The value itself? Or the time at which it happens? Surely, it depends on the initial conditions since different trajectories of ODE intersect nullcline $dot{M} = 0$ at different points. You can get just any of them by starting close to it.
$endgroup$
– Evgeny
Jan 27 at 18:27
$begingroup$
You are correct. I tried solving this problem by linking the initial conditions to the maximum via taylor series, but that results in an intractable polynomial-exponential mix (I also had to expand to 6 terms). I just want a simple expression for the max :(
$endgroup$
– Mike Flynn
Jan 28 at 19:17
$begingroup$
You are correct. I tried solving this problem by linking the initial conditions to the maximum via taylor series, but that results in an intractable polynomial-exponential mix (I also had to expand to 6 terms). I just want a simple expression for the max :(
$endgroup$
– Mike Flynn
Jan 28 at 19:17
$begingroup$
I doubt that it is possible to find such expression, but I might be wrong... At least I don't see any good idea for how this could be find without numerical tabulation or solving the ODE. I see a sort of workaround for your question if you consider $T(0)$ and "time $t^ast$ to reach maximum $M$" as your initial data. In that case you solve for $M$ in equation $dot{M} = 0$ substituting $T$ with solution to $dot{T} = D - gamma T$ at $t = t^ast$ with known $T(0)$. But I'm not sure that this is what you need.
$endgroup$
– Evgeny
Jan 28 at 20:55
$begingroup$
I doubt that it is possible to find such expression, but I might be wrong... At least I don't see any good idea for how this could be find without numerical tabulation or solving the ODE. I see a sort of workaround for your question if you consider $T(0)$ and "time $t^ast$ to reach maximum $M$" as your initial data. In that case you solve for $M$ in equation $dot{M} = 0$ substituting $T$ with solution to $dot{T} = D - gamma T$ at $t = t^ast$ with known $T(0)$. But I'm not sure that this is what you need.
$endgroup$
– Evgeny
Jan 28 at 20:55
add a comment |
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$begingroup$
So what kind of analysis of the maximum do you want? The value itself? Or the time at which it happens? Surely, it depends on the initial conditions since different trajectories of ODE intersect nullcline $dot{M} = 0$ at different points. You can get just any of them by starting close to it.
$endgroup$
– Evgeny
Jan 27 at 18:27
$begingroup$
You are correct. I tried solving this problem by linking the initial conditions to the maximum via taylor series, but that results in an intractable polynomial-exponential mix (I also had to expand to 6 terms). I just want a simple expression for the max :(
$endgroup$
– Mike Flynn
Jan 28 at 19:17
$begingroup$
I doubt that it is possible to find such expression, but I might be wrong... At least I don't see any good idea for how this could be find without numerical tabulation or solving the ODE. I see a sort of workaround for your question if you consider $T(0)$ and "time $t^ast$ to reach maximum $M$" as your initial data. In that case you solve for $M$ in equation $dot{M} = 0$ substituting $T$ with solution to $dot{T} = D - gamma T$ at $t = t^ast$ with known $T(0)$. But I'm not sure that this is what you need.
$endgroup$
– Evgeny
Jan 28 at 20:55