Independent Mean/Variance Regulation in the Self-Enhanced Degradation MotifChris Rackauckas, Likun Zheng, Julian Sosnik, Tom Schilling, Qing Nie

Phenomenological Model

The Zebrafish Hindbrain• The hindbrain is segmented due to spatial patterning of the Hox genes.

We wish to model the effects of the cellular retinoic acid binding proteins (crabps)
and Cyp26a1 on the noise of the RA gradiant in zebrafish hindbrain development.

• This portion of the brain, known as the “hindbrain“ or the “brainstem“, collectivelyregulates vital bodily processes such as breathing, swallowing, blood circulation,and muscle tone.

• Assume the regulation of
RA is Michaelis-Menton.

• Signals produced by mesoderm lying posterior to the hindbrain include RA, andFGF, which with the help of the HOX genes help divide the hindbrain into rhombomeres.

• Let η, the basal rate of RA
degradation, be a ”small
but not insignificant” constant.

• We wish to understand how noise contributes and is regulated within this spatial
patterning network.

Conclusions• We assume the basal rate of RA
degradation is ”small but not insignificant” and the binding rate ofRA to RA-R is large.
• Knockdown of Cyp26a1:– The mean increases.
– The variance is unchanged.

• Let [Crabp] effect thebinding of RA to RA-R andof Cyp26a1 to RA linearly.

• Knockdown of Crabp2a:– The mean is unchanged.

• Let [Cyp]max be themaximum
upreglated
Cyp26 due to RA-R.

d[RA]=– The variance increases.

!

α0 [Cyp]max [RA − R][β −

ω0 /[Crabp] + ω1 + [RA − R]

+ γ0 [Crabp] + γb + η[RA]+δ[RA − R]]dt + σdWt

Future Directions• Incorporate additional effects into the model.

d[RA − R]=((|γ0 [Crabp] + γb )[RA] − δ[RA − R]) dt

d[Crabp]=0– RA upregulates Crabp2a.
– There limited amounts of receptors.

The Fluctuation-Dissipation Theorem• Investigate the effects of Cyp26b1 and Crabp2b proteins.

We can write a system of stochastic differential equations as• Examine the regulatory effects due to transcriptional delays mathematically.

dX = µ(X, t)dt + Γ(X, t)dWt ,• Determine the color of the noise.

Take the Jacobian of the deterministic system around a steady state Xss to get– Mathematically determine the approximate color of the noise.
dX = Jµ (Xss , t)dt + Γ(Xss , t)dWt .

The Interaction Network

The Fluctuation-Dissipation Theorem states that the variance-covariance matrix, Σ, canbe found using the formulaT

2Jµ (Xss , t)Σ(Xss , t) + Σ(Xss , t)Jµ (Xss , t) = −Γ (Xss , t).
• Cells unleash RA from nutrientsin the yolk.
• Cyp26a1 is an intracellular(non-diffusive) molecule which
degrades RA.
• Receptor-bound RA (RA-R)
upregulates the productionof Cyp26a1, creating a selfenhanced degradation loop.
• Cellular retonoic acid binding
proteins (Crabps) can both help
deliver RA to its receptors aswell as deliver it to Cyp26s
for degradation. One of these,
Crabp2a, is also induced by RA
and can increase self-enhanceddegradation.

– Investigate whether the color of the noise holds any information such as
temporal delays.
– Develop an experimental method in order to accurately measure noisefrequencies.
• Examine the spatial properties of the noise.
– Extend the model to a spatial dimension.

Solution– Perturb the spatial noise using mosaic spatial misexpression experiments.

WLOG, δ = 1. Define the function:
22σ[RA] (α, β, ω, η, δ, γ) ≈ σ f (α, β, η, ω, γ)• Look into the transition events in hindbrain development.
– Examine both mathematically and experimentally critical transition behaviors through the autocorrelation lags, variance changes, ”flickering”,spatial coherence.

∂f
Since during Crabp2a knockdown α  1, ∂ω≈ 0.
Let the total change from a knockdown be defined as∆ζ f = | lim f − lim f |.
ζ→∞

∆α f =2(1 + η)
4η(γ + 1 + η)

ζ→0=

11+η2η γ + 1 + η≈0if γ  η1 during Cyp knockdown. And since α  1 during Crab knockdown,∆γ f ≈
which is large if η  1.

12ηAcknowledgments
This investigation was supported by the National Institute of Biomedical Imaging and Bioengineering, National Research Service Award EB009418 from the University of California,Irvine, Center for Complex Biological Systems. I would like to thank Professor Qing Nie forhis mentorship and Dr. Julian Sosnik for his time and effort.