Parameter geography of a two-site phosphorylation systemKee Myoung Nam1 and Jeremy Gunawardena2

1 Swarthmore College, Swarthmore, PA 19081. 2 Department of Systems Biology, Harvard Medical School, Boston, MA 02115.

Abstract

The steady-state dynamics of a multi-site phosphorylation system may berealized as a polynomial system whose parameters are combinations of rateconstants, and whose dynamical variables are concentrations of varioussubstrate/enzyme species. However, the difficulty of numerically estimatingthese rate constants and the sheer complexity of the system poseinsurmountable problems to traditional algebraic geometric methods such asGrobner bases. Instead, we adopt recently developed methods fromnumerical algebraic geometry to efficiently solve for the steady-states of thissystem at numerous parameter values, thereby mapping the “parametergeography” of the system and identifying which regions of the parameterspace give rise to multistability. We pursue this analysis for a two-sitephosphorylation system under assumptions of distributivity, sequentiality, anda physiologically realistic enzymology.

System model

Consider a substrate S0 with two phosphorylation sites. Both sites arephosphorylated by a kinase E and de-phosphorylated by a phosphatase F .

Assuming a physiologically realistic enzymology, we get four reactions andthree conservation laws:

S0 + Ek1−⇀↽−k2

Y1k3−→ Y2

k4−⇀↽−k5

S1 + E

S1 + Ek6−⇀↽−k7

Y3k8−→ Y4

k9−⇀↽−k10

S2 + E

S2 + Fk11−⇀↽−k12

Z4k13−→ Z3

k14−⇀↽−k15

S1 + F

S1 + Fk16−⇀↽−k17

Z2k18−→ Z1

k19−⇀↽−k20

S0 + F

Stot =2∑

i=0

[Si] +4∑

j=1

[Yj] +4∑

k=1

[Zk ] ; Etot = [E ] +4∑

i=1

[Yi] ; Ftot = [F ] +4∑

i=1

[Zi]

on which we can apply the law of mass action with the rate constants ki fori = 1, . . . , 20 to construct a nonlinear differential system in 13 variables.

Applying the linear framework

We apply the linear framework, developed by Prof. Jeremy Gunawardena, todownsize and de-dimensionalize the system into a polynomial system in twovariables u = [E ]

Etotand v = [F ]

Ftot:

1u= 1 +

Stot

Etot

ψ∗2

ψ1 + ψ∗2u + ψ∗

3v1v= 1 +

Stot

Ftot

ψ∗3

ψ1 + ψ∗2u + ψ∗

3vwith ψ1, ψ

∗2, ψ

∗3 defined as follows:

ψ1 = 1 +α

ζ

vu+ βζ

uv

ψ∗2 =

αϵ0ζ

vu+ ϵ1 + βϵ2ζ

uv

ψ∗3 =

αϕ0

ζ

vu+ ϕ1 + βϕ2ζ

uv

where ζ = EtotFtot

and α, β, ϵ0, ϵ1, ϵ2, ϕ0, ϕ1, ϕ2 are various combinations of theconserved quantities Stot,Etot,Ftot and the rate constants ki for i = 1, . . . , 20.These latter eight parameters comprise the standard basis for theeight-dimensional parameter space of the system.

Homotopy continuation, Bertini, and Paramotopy

Consider a polynomial system f(z) = 0, where z ∈ Cn. This system can besolved via homotopy continuation:1. Build and solve a start system g(z) with known solutions;2. Construct a homotopy H(z, t) : Cn × [0,1] → Cn such that H(z,1) = g(z)

and H(z,0) = f(z);3. Follow the paths given by the homotopy from t = 1 to t = 0.Bertini is a newly developed program that applies this method to solvingpolynomial systems. Paramotopy uses Bertini as an engine to solvepolynomial systems for many sets of parameters efficiently.

Positive measure

Assume we have a multistable region R in eight-dimensional parameterspace. Assume that, by random sampling of points, we find a point p in R.Then we know that, with probability one (i.e., almost surely), p lies in asubset of R with positive measure.

A first look at shape and connectedness

We restricted our view of parameter space to the hypercube with all eightparameters varying from 0.1 to 10. Henceforth we refer to this region as the10-cube.

Figure : Two projections of a subset of the multistable region at Stot/Etot = Stot/Ftot = 10 in the10-cube, generated by iteratively sampling under the normal distribution generated by themultistable points. These plots suggest that the multistable region contains one “denselyconnected” subregion within the given boundaries.

Convexity

Intersecting the multistable region in the 10-cube with random lines gives riseto lines with multiple intervals of intersection. This implies non-convexity.

Volume

Lower Stot* Low Stot Mid Stot High Stot

!0,!1,!2

"0, "1, "2

"0,#,$

!0,#,$* Stot has been chosen such that the parameter region is completely monostable.

2

Figure : Effect of increasing Stot in a small hyper-solid in parameter space. Green indicatesmultistability; black, monostability.

Upon further study, we theorize that the volume of the multistable region:▶ Reaches zero at sufficiently low Stot;▶ Saturates (i.e., stops growing) at sufficiently high Stot;▶ Forms a sigmoidal curve with increasing Stot.

Figure : Growth of multistable region in the 10-cube as a function of Stot, assuming Etotand Ftot are constant.

Current conclusions

▶ There exists a subset of the multistable region with positive measure.▶ The multistable region grows in volume, both locally and globally, as the

ratios StotEtot

and StotFtot

increase. This growth follows a sigmoidal curve.▶ The multistable region contains one densely connected subregion, which

implies there exists a low upper bound on the probability that a multistablepoint lies outside this subregion.

▶ The multistable region is not convex.

Acknowledgments

I would like to thank Prof. Jeremy Gunawardena for guiding the project to itscurrent state of completion; Prof. Dan Bates and Dan Brake of ColoradoState University for providing their expertise on the theory and machinerybehind Bertini and Paramotopy; Benjamin Gyori of the National University ofSingapore for writing sections of the code used for sampling andvisualization; and all of the above for conducting many helpful discussions.

This research was generously supported by the Harvard FAS Center for Systems Biology and the Gwill York and Paul Maeder Research Award for Systems Biology. Date: August 15th, 2013.