Derivation of exponential decay equations for mRNA, PrPC and PrPSc.

prpsc-diffeq-derivation

mRNA
====

R(0) = R0
R(infty) = R1

R(t) = R1 + (R0 - R1)exp(-lambda*t)

PrPC
====

S'(t) = alpha*R(t) - mu*S(t)
S(0) = S0
S(infty) = S1
steady state so
S'(0) = 0 = alpha*R0 - mu*S0
alpha*R0 = mu*S0
alpha = mu*S0/R0

substitute in R(t)
S'(t) = alpha*(R1 + (R0-R1)exp(-lambda*t)) - mu*S(t)
S'(t) = alpha*R1 + alpha*(R0-R1)exp(-lambda*t) - mu*S(t)
guess S(t) = A + B*exp(-lambda*t) + C*exp(-mu*t)
differentiate S'(t) = -lambda*B*exp(-lambda*t) + -mu*C*exp(-mu*t)
set S'(t) = S'(t)
alpha*R1 + alpha*(R0-R1)exp(-lambda*t) - mu*S(t) = -lambda*B*exp(-lambda*t) + -mu*C*exp(-mu*t)
recall S(t) = A + B*exp(-lambda*t) + C*exp(-mu*t)
alpha*R1 + alpha*(R0-R1)exp(-lambda*t) - mu*[A + B*exp(-lambda*t) + C*exp(-mu*t)] = -lambda*B*exp(-lambda*t) + -mu*C*exp(-mu*t)
(alpha*R1 - mu*A) + (alpha*(R0-R1) - mu*B + lambda*B)*exp(-lambda*t) + (-mu*C + mu*C)*exp(-mu*t) = 0

# A
alpha*R1 - mu*A = 0
alpha*R1 = mu*A
A = alpha*R1 / mu
alpha = mu*S0/R0
A = S0*R1/R0

# B
alpha*(R0-R1) - mu*B + lambda*B = 0
lambda*B - mu*B = -alpha*(R0 - R1)
B = -alpha*(R0 - R1) / (lambda - mu)
B = alpha*(R0 - R1) / (mu - lambda)
alpha = mu*S0/R0
B = mu*(S0/R0)*(R0-R1) / (mu-lambda)
B = S0*(1-R1/R0) / (1-lambda/mu)


# C
(-mu*C + mu*C)*exp(-mu*t) = 0
0 = 0

S(0) = S0 = A + B*exp(-lambda*0) + C*exp(-mu*0)
S0 = A + B + C
C = S0 - A - B
C = S0 - S0*R1/R0 - S0*(1-R1/R0) / (1-lambda/mu)
C = S0( 1 - R1/R0 - (1-R1/R0)/(1-lambda/mu) )
C = S0( 1 - R1/R0 + (R1/R0 - 1)/(1-lambda/mu) )
S(infty) = S1 = A + B*0 + C*0 = A

A = alpha*R1 / mu = S1 
S1 = R1 * (alpha/mu)
alpha = mu*S0/R0
S1 = R1 * S0/R0
S1 = S0 * R1/R0 # ratio of knockdown is same

S(t) = A + B*exp(-lambda*t) + C*exp(-mu*t)
S(t) = S0*R1/R0 + (S0*(1-R1/R0) / (1-lambda/mu))*exp(-lambda*t) + S0*( 1 - R1/R0 + (R1/R0 - 1)/(1-lambda/mu) )*exp(-mu*t)

special case for mu=lambda:
S(t) = S0 * R1/R0 + S0 * (1-R1/R0) * (1 + mu * t) * exp(-mu*t)


# go back and check that derivative is right
S(t) = S0*R1/R0 + (S0*(1-R1/R0) / (1-lambda/mu))*exp(-lambda*t) + S0*( 1 - R1/R0 + (R1/R0 - 1)/(1-lambda/mu) )*exp(-mu*t)
S'(t) = -lambda*(S0*(1-R1/R0) / (1-lambda/mu))*exp(-lambda*t) + -mu*S0*( 1 - R1/R0 + (R1/R0 - 1)/(1-lambda/mu) )*exp(-mu*t)


PrPSc
====

T'(t) = beta*S(t) - v*T(t)
T(0) = T0
T(infty) = T1
steady state so
T'(0) = 0 = beta*S0 - V*T0
beta*S0 = v*T0
beta = v*T0/S0

T(t) = A + B*exp(-lambda*t) + C*exp(-mu*t) + D*exp(-v*t)
T'(t) = -lambda*B*exp(-lambda*t) - mu*C*exp(-mu*t) - v*D*exp(-v*t)
and also
T'(t) = beta*S(t) - v*[A + B*exp(-lambda*t) + C*exp(-mu*t) + D*exp(-v*t)]
therefore
beta*S(t) - v*[A + B*exp(-lambda*t) + C*exp(-mu*t) + D*exp(-v*t)] = -lambda*B*exp(-lambda*t) - mu*C*exp(-mu*t) - v*D*exp(-v*t)
beta*[S0*R1/R0 + (S0*(1-R1/R0) / (1-lambda/mu))*exp(-lambda*t) + S0*( 1 - R1/R0 + (R1/R0 - 1)/(1-lambda/mu) )*exp(-mu*t)] - v*[A + B*exp(-lambda*t) + C*exp(-mu*t) + D*exp(-v*t)] = -lambda*B*exp(-lambda*t) - mu*C*exp(-mu*t) - v*D*exp(-v*t)

# A = Constant terms
beta*S0*R1/R0 - v*A = 0
A = beta*S0*(R1/R0) / v
beta = v*T0/S0
A = v*(T0/S0)*S0*(R1/R0) / v
A = T0*R1/R0 # knockdown level is same

# B = exp(-lambda*t) terms
beta*(S0*(1-R1/R0) / (1-lambda/mu))*exp(-lambda*t) - v*B*exp(-lambda*t) = -lambda*B*exp(-lambda*t)
beta*(S0*(1-R1/R0) / (1-lambda/mu))*exp(-lambda*t) + lambda*B*exp(-lambda*t) - v*B*exp(-lambda*t) = 0
beta*(S0*(1-R1/R0) / (1-lambda/mu)) + lambda*B - v*B = 0
beta*(S0*(1-R1/R0) / (1-lambda/mu)) = v*B - lambda*B
B = beta*(S0*(1-R1/R0) / (1-lambda/mu))/(v - lambda)
B = v*(T0/S0)*(S0*(1-R1/R0) / (1-lambda/mu))/(v - lambda)

# C = exp(-mu*t) terms
beta*S0*( 1 - R1/R0 + (R1/R0 - 1)/(1-lambda/mu) )*exp(-mu*t) - v*C*exp(-mu*t) = -mu*C*exp(-mu*t)
beta*S0*( 1 - R1/R0 + (R1/R0 - 1)/(1-lambda/mu) )*exp(-mu*t) + mu*C*exp(-mu*t) - v*C*exp(-mu*t) = 0
beta*S0*( 1 - R1/R0 + (R1/R0 - 1)/(1-lambda/mu) ) + mu*C - v*C = 0
beta*S0*( 1 - R1/R0 + (R1/R0 - 1)/(1-lambda/mu) ) = v*C - mu*C
C = beta*S0*( 1 - R1/R0 + (R1/R0 - 1)/(1-lambda/mu) ) / (v - mu)
beta = v*T0/S0
C = v*T0*( 1 - R1/R0 + (R1/R0 - 1)/(1-lambda/mu) ) / (v - mu)

# D = exp(-v*t) terms
-v*D*exp(-v*t) = -v*D*exp(-v*t)
again, no information about D
T(infty) = A = T0*R1/R0 # knockdown level is same
T(0) = T0 = A + B + C + D
D = T0 - A - B - C
D = T0 - (T0*R1/R0) - (v*(T0/S0)*(S0*(1-R1/R0) / (1-lambda/mu))/(v - lambda)) - (v*T0*( 1 - R1/R0 + (R1/R0 - 1)/(1-lambda/mu) ) / (v - mu))

T(t) = T0*R1/R0 + (v*(T0/S0)*(S0*(1-R1/R0) / (1-lambda/mu))/(v - lambda))*exp(-lambda*t) + (v*T0*( 1 - R1/R0 + (R1/R0 - 1)/(1-lambda/mu) ) / (v - mu))*exp(-mu*t) + (T0 - (T0*R1/R0) - (v*(T0/S0)*(S0*(1-R1/R0) / (1-lambda/mu))/(v - lambda)) - (v*T0*( 1 - R1/R0 + (R1/R0 - 1)/(1-lambda/mu) ) / (v - mu)))*exp(-v*t)

looks like you dropped a beta at line 115.

big derivations like this are easier to do on a whiteboard and then transcribe to text -- the text format makes it very hard to spot things like that extra beta hanging out in front of the brackets. of course, then you have to be careful not to make transcription errors (as I did with R0 instead of R1 in the expression for A).

Incredible that you spotted that.
Yes, fixed and plugged into the simulation and now it works.