Tuesday 07/30/13

9:30 A – 5:00 P, 9:00 P- 12:00 P

STORM

  • set up O/N Ph-wt Pc-cy7

Modeling

Using renewal equations which say fano factor of time is eta^2 of transcript count variability (covN) should go like 1/kf for all N for kb=0. This should be obvious, if kb is zero, the equilibrium condition is just to sit in the N-1 state and fire at rate kf,

  • for kf > kb, as N-> inf: fanoT = covN -> kf+kb / (kf-kb)^2
  • for kf < kb, as N-> inf covN -> inf

Variable defs:
M =

[ – alpha – kf, kf, 0, 0, alpha]
[ kb, – kb – kf, kf, 0, 0]
[ 0, kb, – kb – kf, kf, 0]
[ 0, 0, kb, – kb – kf, kf]
[ gamma, 0, 0, kb, – gamma – kb]

For N=3, kb=0, alpha = 0; mu = 1000/(d((3gamma)/kf + 1)),

in fact,

For N=3, kb=0,

kt/(d((4gammakf^3)/(kf^4 + alphakf^3) + 1))

Which for alpha = 0;
mu = kt/(N((3gamma)/kf + 1)),

so for gamma = kf, kf has no effect on gamma. (You leave the synthesizing state faster and you enter it faster so no net effect).
To keep mu fixed in the case where kb = 0, we need to divide delta by Ns, since we have D/Ns in the denominator.

var is not independent of kf however. For alpha=kb=0 and gamma=kf, Ns = 5:
kt/(5d) – kt^2/(25d^2) + (kt^2(d^4 + 4d^3kf + 6d^2kf^2 + 4dkf^3 + kf^4))/(5d(d^5 + 5d^4kf + 10d^3kf^2 + 10d^2kf^3 + 5d*kf^4))

Without adjusting kf, the system spends more time in the off state as N is increased, and so the coefficient of variation increases by becoming increasingly bursty (in the Pedraza model, bursty-ness is controlled independently of N by fiat).

The fully symmetric, fully reversible case breaks the gestation intuition since the system becomes instead just a random walk with one lucky state. Increasing the number of (non-lucky) states just decreases the probability of intersecting the lucky state.

IMPORTANT: Sanchez equations require reversibility, these zero limits are probably not valid.

Illustration:

M = [-kf, kf, 0
kb, -kb-kf kf
0, kb , -kb];

subs(mu,kb,0) = kt/d % This is correct, we expect just Poisson(kt/d) when kb -> 0.
subs(sigma2,kb,0) = kt/d – kt^2/d^2 + (kt^2(d^2 + 2dkf + kf^2))/(d(d^3 + 2d^2kf + d*kf^2)) % This is NOT Poisson(kt/d), var = kt/d.

This entry was posted in Summaries, Transcription Modeling. Bookmark the permalink.