Metastable Regimes and Tipping points of Biochemical Networks with - - PowerPoint PPT Presentation

Metastable Regimes and Tipping points of Biochemical Networks with Potential Applications in Precision Medicine

Satya S. Samal Joint Research Center for Computa7onal Biomedicine (JRC-COMBINE) RWTH Aachen University 5th July, Bonn

Jointly with, Jeyashree Krishnan, Ali Hadizadeh Esfahani Christoph Lüders, Andreas Weber, and Ovidiu Radulescu

Tipping points / Cri-cal transi-ons

• Disease: Devia&on of a few system parameters from

‘normal’ state qualita&vely affec&ng system behavior

MAPK cascade : Huang and Ferrell 96

p healthy disease Risk zone

• rder parameter

System response

• Sudden change in a dynamical system's state

§ Bifurca&ons, Phase Transi&ons, … § Can be predic&ve

Chen et al. (2012). Dakoset al. (2012). Schefferet al. (2009).

Precision medicine

• Predict therapy outcome (at individual/micro-segments).
• Extrapola<on of mathema<cal models.
• Heterogeneity of pa<ents.
• Pa<ent specificity parameters in models.
• Non-sta<onary <me series.
• Non constancy of underlying biological mechanism due to (clinical/biological)

perturba<ons.

• Altera<ons in signalling pathways (such as MAPK/PI3K).
• Pathway redundancy and mul<ple feedback regula<on are obstacles against

cancer targeted therapies.

Metastable regimes

• Trajectories of ODEs consist of transi0ons

between slow regions.

• Slow regions are denoted by low dimensional

manifolds are called metastable regimes.

• In
• ur

work, the metastable regimes correspond to tropical equilibra0on (TE) solu0ons.

ØSlowness follows from the compensa0on

• f

dominant monomials. ØTropical geometry provides a framework to compute such states.

Crazy quilt: Representation of MRs. Dominant vector fields (red arrows) confine the trajectory to low dimensional patches on which act weak uncompensated vector fields (blue arrows).

Tropical equilibra.on solu.ons

• Order analysis of ODEs

Ø ̇ "1 = −"16 +"13"2 −"13 +"1"22

Ø Order of the variables Ø "1 = "1εa1, "2 = "2εa2 Ø Order of the monomials

Ø"16 ="16ε6a1 Ø"13"2="13"2ε3a1+a2 Ø"13="13ε3a1 Ø"1"22="1"22εa1+2a2

• Metastable regimes for x1 andx2 results into following

equilibraCon condiCon

Ø min(3a1+a2 a1+2a2) = min(6a1,3a1)

• Branches: Equivalence classes of TE soluCons.

Ø(Closure) for each branch there exists a unique convex polytope.

• Minimal branch: Branch corresponding to maximal

polytope with respect to inclusion.

Branches

• f

tropical solutions correspond to half lines (orthogonal to the thick edges of newton polytope) and are given by a1 = a2 >= 0, a1 <=0, a2 = 5/2 a1 .

Tropical minimal branches

Tropical equilibra.ons (TE) are solu.on of the below for orders a Rescaled system

Sensi&ve/Robust parameters

Full distance for the j parameter Distance for the j parameter along Xmvariable. Networks Models (along with kine>c rate parameters) Compute tropical equilibrations (minimal branches) Vary parameters and compute their effect on the minimal branches

Samal et al. (submiEed)

Results: Network model

Hatakeyama, M. et al (2013) Biomodels: BIOMD0000000146

• The

model was mo4vated by experimental works on the Heregulin s4mulated ErbB receptor and demonstrates the Akt-induced inhibi4on of the MAPK pathway via phosphoryla4on of Raf-1.

• This CRN model has 33 species and 34
• reac4ons. 21 reac4ons have Michaelis-

Menten kine4cs and 12 have mass ac4on kine4c laws.

Results: Distances

Robust parameters Sensitive parameters Normalised distances for select parameters. D1 full distance, D2 distance along MAPKPP axis, D3 distance along AKTPiPP axis. DistribuCon of D1 distances.

Change in protein data concentrations

• Quan%fy the overlap with tropical sensi%vity scores:
• Area under the curve (AUC):
• Breast cancer data:0.55
• Skin Cutaneous Melanoma: 0.85
• Breast cancer subtypes: (average): 0.72

*0.50 is random guessing. Sta%s%cal comparison of disease versus healthy samples from BRCA and SKCM cancers. A low p-value suggests significant change in rela%ve concentra%ons.

So#ware

• PtCut - Check Christoph Lüders’s webpage: h9p://wrogn.com/ptcut

Future work (SYMBIONT)

• Compute robustness for different models.
• Multi-parameter perturbation (efficient sampling techniques).
• Better distance measures.
• Parametric solving
• Without fixing the parameter orders.
• Learn parameters from experimental (noisy) data.
• Tropical interpolation.
• Integrate disease data/networks.
• Do we have the “executable” models ?

Conclusions

• A “global” method to test robustness of biochemical reaction

networks.

• Dependence of change in tropical solutions w.r.t. model parameters.
• A potential in-silico tool to identify putative drug targets.
• Provides a ranked list of targets for biologists for experimental validation.
• Comparison with protein concentration data.
• Cancer vs Healthy data.
• Future outlook in context of SYMBIONT.

Thank you