Parameter geography of a two-site phosphorylation...
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Parameter geography of a two-site phosphorylation system Kee Myoung Nam 1 and Jeremy Gunawardena 2 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 be realized as a polynomial system whose parameters are combinations of rate constants, and whose dynamical variables are concentrations of various substrate/enzyme species. However, the difficulty of numerically estimating these rate constants and the sheer complexity of the system pose insurmountable problems to traditional algebraic geometric methods such as Gr¨ obner bases. Instead, we adopt recently developed methods from numerical algebraic geometry to efficiently solve for the steady-states of this system at numerous parameter values, thereby mapping the “parameter geography” of the system and identifying which regions of the parameter space give rise to multistability. We pursue this analysis for a two-site phosphorylation system under assumptions of distributivity, sequentiality, and a physiologically realistic enzymology. System model Consider a substrate S 0 with two phosphorylation sites. Both sites are phosphorylated by a kinase E and de-phosphorylated by a phosphatase F . Assuming a physiologically realistic enzymology, we get four reactions and three conservation laws: S 0 + E k 1 - ⇀ ↽ - k 2 Y 1 k 3 -→ Y 2 k 4 - ⇀ ↽ - k 5 S 1 + E S 1 + E k 6 - ⇀ ↽ - k 7 Y 3 k 8 -→ Y 4 k 9 - ⇀ ↽ - k 10 S 2 + E S 2 + F k 11 - ⇀ ↽ - k 12 Z 4 k 13 -→ Z 3 k 14 - ⇀ ↽ - k 15 S 1 + F S 1 + F k 16 - ⇀ ↽ - k 17 Z 2 k 18 -→ Z 1 k 19 - ⇀ ↽ - k 20 S 0 + F S tot = 2 ∑ i =0 [S i ]+ 4 ∑ j =1 [Y j ]+ 4 ∑ k =1 [Z k ]; E tot =[E ]+ 4 ∑ i =1 [Y i ]; F tot =[F ]+ 4 ∑ i =1 [Z i ] on which we can apply the law of mass action with the rate constants k i for i = 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, to downsize and de-dimensionalize the system into a polynomial system in two variables u = [E ] E tot and v = [F ] F tot : 1 u = 1 + S tot E tot ψ * 2 ψ 1 + ψ * 2 u + ψ * 3 v 1 v = 1 + S tot F tot ψ * 3 ψ 1 + ψ * 2 u + ψ * 3 v with ψ 1 ,ψ * 2 ,ψ * 3 defined as follows: ψ 1 = 1 + α ζ v u + βζ u v ψ * 2 = αϵ 0 ζ v u + ϵ 1 + βϵ 2 ζ u v ψ * 3 = αϕ 0 ζ v u + ϕ 1 + βϕ 2 ζ u v where ζ = E tot F tot and α,β,ϵ 0 ,ϵ 1 ,ϵ 2 ,ϕ 0 ,ϕ 1 ,ϕ 2 are various combinations of the conserved quantities S tot , E tot , F tot and the rate constants k i for i = 1,..., 20. These latter eight parameters comprise the standard basis for the eight-dimensional parameter space of the system. Homotopy continuation, Bertini, and Paramotopy Consider a polynomial system f(z)= 0, where z ∈ C n . This system can be solved via homotopy continuation: 1. Build and solve a start system g(z) with known solutions; 2. Construct a homotopy H (z, t ): C n × [0, 1] → C n 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 solving polynomial systems. Paramotopy uses Bertini as an engine to solve polynomial systems for many sets of parameters efficiently. Positive measure Assume we have a multistable region R in eight-dimensional parameter space. 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 a subset of R with positive measure. A first look at shape and connectedness We restricted our view of parameter space to the hypercube with all eight parameters varying from 0.1 to 10. Henceforth we refer to this region as the 10-cube. Figure : Two projections of a subset of the multistable region at S tot /E tot = S tot /F tot = 10 in the 10-cube, generated by iteratively sampling under the normal distribution generated by the multistable points. These plots suggest that the multistable region contains one “densely connected” subregion within the given boundaries. Convexity Intersecting the multistable region in the 10-cube with random lines gives rise to lines with multiple intervals of intersection. This implies non-convexity. Volume Lower S tot * Low S tot Mid S tot High S tot * S tot has been chosen such that the parameter region is completely monostable. Figure : Effect of increasing S tot in a small hyper-solid in parameter space. Green indicates multistability; black, monostability. Upon further study, we theorize that the volume of the multistable region: ▶ Reaches zero at sufficiently low S tot ; ▶ Saturates (i.e., stops growing) at sufficiently high S tot ; ▶ Forms a sigmoidal curve with increasing S tot . Figure : Growth of multistable region in the 10-cube as a function of S tot , assuming E tot and F tot 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 S tot E tot and S tot F tot 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 multistable point lies outside this subregion. ▶ The multistable region is not convex. Acknowledgments I would like to thank Prof. Jeremy Gunawardena for guiding the project to its current state of completion; Prof. Dan Bates and Dan Brake of Colorado State University for providing their expertise on the theory and machinery behind Bertini and Paramotopy; Benjamin Gyori of the National University of Singapore for writing sections of the code used for sampling and visualization; 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.
Transcript of Parameter geography of a two-site phosphorylation...
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.
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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.
