RNA flux explorer

Compare control and case

Rates

Drag a dial up or down, scroll over it, or use arrow keys. Turn it all the way down (or type 0) to switch a step off. Hold Shift for fine control. Double-click to reset.

Example transcripts

Scenario Switch transcription on from zero Shut transcription off from steady state Change from control rates to case rates, starting at the control steady state

Plot relative to steady state

Each dot = molecules

At this moment

At steady state

Abundance

Molecules per cell

Share of cellular RNA

Time course

The model and its solution

N is nuclear RNA and C is cytoplasmic RNA, in molecules per cell. Time t is in minutes.

Equations

dN/dt = s − (k_exp + k_nd)·N, dC/dt = k_exp·N − k_cd·C

It helps to name the two rates that set the timescales: a = k_exp + k_nd, the total rate at which RNA leaves the nucleus, and b = k_cd, the rate at which it leaves the cytoplasm.

Steady state

N* = s/a, C* = k_exp·N*/b

The nuclear fraction at steady state, N*/(N*+C*) = b/(b + k_exp), does not depend on synthesis or nuclear decay. The fraction of transcripts exported before they decay is k_exp/a.

Solution over time

Starting from any levels N₀, C₀ with constant rates:

N(t) = N* + (N₀ − N*)·e^(−at)

C(t) = C* + (C₀ − C*)·e^(−bt) + k_exp(N₀ − N*)/(b − a)·(e^(−at) − e^(−bt))

Nuclear RNA relaxes as a single exponential at rate a. Cytoplasmic RNA mixes two exponentials, because it is fed by the nucleus while relaxing at its own rate b. The app evaluates this exact solution directly.

Special cases: if a = b, replace (e^(−at) − e^(−bt))/(b − a) with t·e^(−at). If a = 0, nothing leaves the nucleus and N(t) = N₀ + s·t grows without limit. If b = 0 and export is on, cytoplasmic RNA grows without limit.

Switching transcription on

Starting from zero (N₀ = C₀ = 0):

N(t) = N*(1 − e^(−at)), C(t) = C*(1 − (b·e^(−at) − a·e^(−bt))/(b − a))

The nucleus reaches half its steady state after ln2/a. The cytoplasm starts slowly, because it must wait for export, and is limited by whichever of a and b is slower.

Shutting transcription off

Setting s = 0 from steady state:

N(t) = N*·e^(−at), C(t) = C*·(b·e^(−at) − a·e^(−bt))/(b − a)

Whole-cell RNA, N(t) + C(t), therefore decays as a sum of two exponentials. A half-life measured from total RNA after a transcription block blends nuclear and cytoplasmic turnover, and it approaches the cytoplasmic half-life only when export is much faster than cytoplasmic decay.

With the current rates

Switching transcription on, t in minutes.