![It is a complex problem
The complete human metabolic network Recon1 [Palsson et al.’07]](https://image.slidesharecdn.com/presentationsocg-220109142515/85/Symposium-of-Computational-Geometry-2021-4-320.jpg)
![Matrix representation of a metabolic network
11 metabolites and 4 reactions
Use S ∈ Rm×n and flux vector v ∈ Rn to express the
change of the mass of each metabolite over time [Palsson’15],
dr
dt
= Sv = S
v1
v2
v3
v4
= v1S(·,1) + v2S(·,2) + v3S(·,3) + v4S(·,4)](https://image.slidesharecdn.com/presentationsocg-220109142515/85/Symposium-of-Computational-Geometry-2021-8-320.jpg)
![Sampling steady states
Sampling could lead to important biological insights! [Palsson’15]
Explore the flux space [Schellenberger,Palsson’09].
We introduce a
Multiphase Monte Carlo Sampling algorithm
based on Billiard Walk](https://image.slidesharecdn.com/presentationsocg-220109142515/85/Symposium-of-Computational-Geometry-2021-11-320.jpg)
![Billiard walk
BW(P, pi ) [Gryazina,Polyak’14]
1 Generate the length of the trajectory L ∼ D.
Existing work on
uniform sampling
1. Hit-and-run
[Lovász et al.’06]
2. Geodesic walk
[Lee et al.’17]
3. Ball walk
[Lee et al.’17]
4. Vaidya walk
[Chen et al.’17]
5. Riemmanian HMC
[Lee et al.’18]
6. Dikin walk
[Laddha et al.’20]
7. Coordinate
Hit-and-Run
[Laddha et al.’21]
8. HMC reflections
[Chevallier et al.’21]](https://image.slidesharecdn.com/presentationsocg-220109142515/85/Symposium-of-Computational-Geometry-2021-12-320.jpg)
![Billiard walk
BW(P, pi ) [Gryazina,Polyak’14]
1 Generate the length of the trajectory L ∼ D.
2 Pick a uniform direction v to define the trajectory.
Existing work on
uniform sampling
1. Hit-and-run
[Lovász et al.’06]
2. Geodesic walk
[Lee et al.’17]
3. Ball walk
[Lee et al.’17]
4. Vaidya walk
[Chen et al.’17]
5. Riemmanian HMC
[Lee et al.’18]
6. Dikin walk
[Laddha et al.’20]
7. Coordinate
Hit-and-Run
[Laddha et al.’21]
8. HMC reflections
[Chevallier et al.’21]](https://image.slidesharecdn.com/presentationsocg-220109142515/85/Symposium-of-Computational-Geometry-2021-13-320.jpg)
![Billiard walk
BW(P, pi ) [Gryazina,Polyak’14]
1 Generate the length of the trajectory L ∼ D.
2 Pick a uniform direction v to define the trajectory.
Existing work on
uniform sampling
1. Hit-and-run
[Lovász et al.’06]
2. Geodesic walk
[Lee et al.’17]
3. Ball walk
[Lee et al.’17]
4. Vaidya walk
[Chen et al.’17]
5. Riemmanian HMC
[Lee et al.’18]
6. Dikin walk
[Laddha et al.’20]
7. Coordinate
Hit-and-Run
[Laddha et al.’21]
8. HMC reflections
[Chevallier et al.’21]](https://image.slidesharecdn.com/presentationsocg-220109142515/85/Symposium-of-Computational-Geometry-2021-14-320.jpg)
![Billiard walk
BW(P, pi ) [Gryazina,Polyak’14]
1 Generate the length of the trajectory L ∼ D.
2 Pick a uniform direction v to define the trajectory.
3 The trajectory reflects on the boundary if necessary.
Existing work on
uniform sampling
1. Hit-and-run
[Lovász et al.’06]
2. Geodesic walk
[Lee et al.’17]
3. Ball walk
[Lee et al.’17]
4. Vaidya walk
[Chen et al.’17]
5. Riemmanian HMC
[Lee et al.’18]
6. Dikin walk
[Laddha et al.’20]
7. Coordinate
Hit-and-Run
[Laddha et al.’21]
8. HMC reflections
[Chevallier et al.’21]](https://image.slidesharecdn.com/presentationsocg-220109142515/85/Symposium-of-Computational-Geometry-2021-15-320.jpg)
![Billiard walk
BW(P, pi ) [Gryazina,Polyak’14]
1 Generate the length of the trajectory L ∼ D.
2 Pick a uniform direction v to define the trajectory.
3 The trajectory reflects on the boundary if necessary.
Existing work on
uniform sampling
1. Hit-and-run
[Lovász et al.’06]
2. Geodesic walk
[Lee et al.’17]
3. Ball walk
[Lee et al.’17]
4. Vaidya walk
[Chen et al.’17]
5. Riemmanian HMC
[Lee et al.’18]
6. Dikin walk
[Laddha et al.’20]
7. Coordinate
Hit-and-Run
[Laddha et al.’21]
8. HMC reflections
[Chevallier et al.’21]](https://image.slidesharecdn.com/presentationsocg-220109142515/85/Symposium-of-Computational-Geometry-2021-16-320.jpg)
![Billiard walk
BW(P, pi ) [Gryazina,Polyak’14]
1 Generate the length of the trajectory L ∼ D.
2 Pick a uniform direction v to define the trajectory.
3 The trajectory reflects on the boundary if necessary.
4 return the end of the trajectory as pi+1.
Existing work on
uniform sampling
1. Hit-and-run
[Lovász et al.’06]
2. Geodesic walk
[Lee et al.’17]
3. Ball walk
[Lee et al.’17]
4. Vaidya walk
[Chen et al.’17]
5. Riemmanian HMC
[Lee et al.’18]
6. Dikin walk
[Laddha et al.’20]
7. Coordinate
Hit-and-Run
[Laddha et al.’21]
8. HMC reflections
[Chevallier et al.’21]](https://image.slidesharecdn.com/presentationsocg-220109142515/85/Symposium-of-Computational-Geometry-2021-17-320.jpg)
![Billiard walk
p
q
pi pi+1
Mixing time: O∗(`2d2) well-rounded convex bodies
[C,Papachristou,Fisikopoulos,Tsigaridas,’21],
Improved cost per point: O(kd) operations [This paper].
`: upper bound on the number of reflections, k: number of facets.](https://image.slidesharecdn.com/presentationsocg-220109142515/85/Symposium-of-Computational-Geometry-2021-18-320.jpg)
![Multiphase Monte Carlo Sampling
MMCS(P0, p, N, set i = 0)
1 Sample O(d) points from Pi with Billiard walk.
2 Estimate the Effective Sample Size (ESS) of the sample
in Pi .
3 Map the sampled points to an isotropic position and
apply the same transformation Ti to Pi , set
Pi+1 = Ti (Pi ).
4 i = i + 1; goto 1.
5 Stop when the sum of ESS N and PSRF 1.1.
Related work
(optimization)
1. [Bertsimas et al.’04]
2. [Kalai et al.’06]](https://image.slidesharecdn.com/presentationsocg-220109142515/85/Symposium-of-Computational-Geometry-2021-20-320.jpg)
![Experiments
MMCS cobra
model (d) Time (sec) (Steps) Time (sec) (Steps)
iAB RBC 283 130 5.20e+01 1.07e+04 7.85e+03 4.05e+08
iAT PLT 636 289 3.25e+02 1.04e+04 1.73e+04 6.68e+08
iML1515 633 4.65e+03 5.65e+04 1.15e+05 3.21e+09
Recon1 931 8.09e+03 1.94e+04 3.20e+05 6.93e+09
Recon2D 2430 2.48e+04 5.44e+04 ∼ 140 days 1.57e+11
Recon3D 5335 1.03e+05 1.44e+05 – –
Package cobra [Palsson, Thiele, Fleming et al.’19]:
state-of-the-art software to analyze metabolic networks,
Coordinate Directions Hit-and-Run [Cousins, Vempala et al.’17].
MMCS outperforms existing software.](https://image.slidesharecdn.com/presentationsocg-220109142515/85/Symposium-of-Computational-Geometry-2021-21-320.jpg)
![Find possible anti-COVID19 targets
Not an anti-viral target Possible anti-viral target
Sample steady states when,
the growth rate of COVID-19 is optimized,
the host biomass production is optimized.
Check if the flux distribution of a reaction changes
[Renz,Widerspick,Dräger’20,’21].](https://image.slidesharecdn.com/presentationsocg-220109142515/85/Symposium-of-Computational-Geometry-2021-23-320.jpg)
Symposium of Computational Geometry 2021
- 1. Geometric Algorithms for Sampling the Flux Space of Metabolic Networks Apostolos Chalkis 1, 2, 3 Vissarion Fisikopoulos (NKUA), Elias Tsigaridas (INRIA), Haris Zafeiropoulos (IMBBC-HCMR) 1National & Kapodistrian U. Athens, Greece 2ATHENA Research & Innovation Center, Greece 3GeomScale org June 11, 2021
- 2. In our cells... We call both the inputs (reactants) and the outputs (products) of a chemical reaction, metabolites. In every cell of our body thousands of chemical reactions are taking place!
- 3. The reactions interact A small fragment of the human metabolic network Q: How can we model all the interactions between chemical sdf reactions in an organism? A: Computational geometry can help!
- 4. It is a complex problem The complete human metabolic network Recon1 [Palsson et al.’07]
- 5. A small example of a metabolic network
- 6. A small example of a metabolic network
- 7. Key Concept of a metabolic network: Reaction Fluxes The i-th reaction has a flux (rate) vi that is flowing. vi multiplies each metabolite in the i-th reaction.
- 8. Matrix representation of a metabolic network 11 metabolites and 4 reactions Use S ∈ Rm×n and flux vector v ∈ Rn to express the change of the mass of each metabolite over time [Palsson’15], dr dt = Sv = S v1 v2 v3 v4 = v1S(·,1) + v2S(·,2) + v3S(·,3) + v4S(·,4)
- 9. Steady states: The network in balance When for each metabolite, the rate of production equals to the rate of consumption, the reactions exactly balance each other. When a flux vector v balances the network, Sv = 0, v is a steady state.
- 10. The region of steady states As a low dimensional polytope. Sv = 0, vlb ≤ v ≤ vub ←→ v=Nx S ∈ Rm×n , v ∈ Rn As a full dimensional polytope P := {x ∈ Rd | Ax ≤ b} A = InN −InN and b = vub vlb N N ∈ Rn×d the matrix of the right nullspace of S.
- 11. Sampling steady states Sampling could lead to important biological insights! [Palsson’15] Explore the flux space [Schellenberger,Palsson’09]. We introduce a Multiphase Monte Carlo Sampling algorithm based on Billiard Walk
- 12. Billiard walk BW(P, pi ) [Gryazina,Polyak’14] 1 Generate the length of the trajectory L ∼ D. Existing work on uniform sampling 1. Hit-and-run [Lovász et al.’06] 2. Geodesic walk [Lee et al.’17] 3. Ball walk [Lee et al.’17] 4. Vaidya walk [Chen et al.’17] 5. Riemmanian HMC [Lee et al.’18] 6. Dikin walk [Laddha et al.’20] 7. Coordinate Hit-and-Run [Laddha et al.’21] 8. HMC reflections [Chevallier et al.’21]
- 13. Billiard walk BW(P, pi ) [Gryazina,Polyak’14] 1 Generate the length of the trajectory L ∼ D. 2 Pick a uniform direction v to define the trajectory. Existing work on uniform sampling 1. Hit-and-run [Lovász et al.’06] 2. Geodesic walk [Lee et al.’17] 3. Ball walk [Lee et al.’17] 4. Vaidya walk [Chen et al.’17] 5. Riemmanian HMC [Lee et al.’18] 6. Dikin walk [Laddha et al.’20] 7. Coordinate Hit-and-Run [Laddha et al.’21] 8. HMC reflections [Chevallier et al.’21]
- 14. Billiard walk BW(P, pi ) [Gryazina,Polyak’14] 1 Generate the length of the trajectory L ∼ D. 2 Pick a uniform direction v to define the trajectory. Existing work on uniform sampling 1. Hit-and-run [Lovász et al.’06] 2. Geodesic walk [Lee et al.’17] 3. Ball walk [Lee et al.’17] 4. Vaidya walk [Chen et al.’17] 5. Riemmanian HMC [Lee et al.’18] 6. Dikin walk [Laddha et al.’20] 7. Coordinate Hit-and-Run [Laddha et al.’21] 8. HMC reflections [Chevallier et al.’21]
- 15. Billiard walk BW(P, pi ) [Gryazina,Polyak’14] 1 Generate the length of the trajectory L ∼ D. 2 Pick a uniform direction v to define the trajectory. 3 The trajectory reflects on the boundary if necessary. Existing work on uniform sampling 1. Hit-and-run [Lovász et al.’06] 2. Geodesic walk [Lee et al.’17] 3. Ball walk [Lee et al.’17] 4. Vaidya walk [Chen et al.’17] 5. Riemmanian HMC [Lee et al.’18] 6. Dikin walk [Laddha et al.’20] 7. Coordinate Hit-and-Run [Laddha et al.’21] 8. HMC reflections [Chevallier et al.’21]
- 16. Billiard walk BW(P, pi ) [Gryazina,Polyak’14] 1 Generate the length of the trajectory L ∼ D. 2 Pick a uniform direction v to define the trajectory. 3 The trajectory reflects on the boundary if necessary. Existing work on uniform sampling 1. Hit-and-run [Lovász et al.’06] 2. Geodesic walk [Lee et al.’17] 3. Ball walk [Lee et al.’17] 4. Vaidya walk [Chen et al.’17] 5. Riemmanian HMC [Lee et al.’18] 6. Dikin walk [Laddha et al.’20] 7. Coordinate Hit-and-Run [Laddha et al.’21] 8. HMC reflections [Chevallier et al.’21]
- 17. Billiard walk BW(P, pi ) [Gryazina,Polyak’14] 1 Generate the length of the trajectory L ∼ D. 2 Pick a uniform direction v to define the trajectory. 3 The trajectory reflects on the boundary if necessary. 4 return the end of the trajectory as pi+1. Existing work on uniform sampling 1. Hit-and-run [Lovász et al.’06] 2. Geodesic walk [Lee et al.’17] 3. Ball walk [Lee et al.’17] 4. Vaidya walk [Chen et al.’17] 5. Riemmanian HMC [Lee et al.’18] 6. Dikin walk [Laddha et al.’20] 7. Coordinate Hit-and-Run [Laddha et al.’21] 8. HMC reflections [Chevallier et al.’21]
- 18. Billiard walk p q pi pi+1 Mixing time: O∗(`2d2) well-rounded convex bodies [C,Papachristou,Fisikopoulos,Tsigaridas,’21], Improved cost per point: O(kd) operations [This paper]. `: upper bound on the number of reflections, k: number of facets.
- 19. Difficulties When the polytope is skinny: The average number of reflections increases. The mixing rate decreases.
- 20. Multiphase Monte Carlo Sampling MMCS(P0, p, N, set i = 0) 1 Sample O(d) points from Pi with Billiard walk. 2 Estimate the Effective Sample Size (ESS) of the sample in Pi . 3 Map the sampled points to an isotropic position and apply the same transformation Ti to Pi , set Pi+1 = Ti (Pi ). 4 i = i + 1; goto 1. 5 Stop when the sum of ESS N and PSRF 1.1. Related work (optimization) 1. [Bertsimas et al.’04] 2. [Kalai et al.’06]
- 21. Experiments MMCS cobra model (d) Time (sec) (Steps) Time (sec) (Steps) iAB RBC 283 130 5.20e+01 1.07e+04 7.85e+03 4.05e+08 iAT PLT 636 289 3.25e+02 1.04e+04 1.73e+04 6.68e+08 iML1515 633 4.65e+03 5.65e+04 1.15e+05 3.21e+09 Recon1 931 8.09e+03 1.94e+04 3.20e+05 6.93e+09 Recon2D 2430 2.48e+04 5.44e+04 ∼ 140 days 1.57e+11 Recon3D 5335 1.03e+05 1.44e+05 – – Package cobra [Palsson, Thiele, Fleming et al.’19]: state-of-the-art software to analyze metabolic networks, Coordinate Directions Hit-and-Run [Cousins, Vempala et al.’17]. MMCS outperforms existing software.
- 22. Estimate flux distributions Reaction GLGNS1 is crucial for the growth rate of an organism. Either slow Human biomass production and small GLGNS1 flux or fast biomass production and large GLGNS1 flux.
- 23. Find possible anti-COVID19 targets Not an anti-viral target Possible anti-viral target Sample steady states when, the growth rate of COVID-19 is optimized, the host biomass production is optimized. Check if the flux distribution of a reaction changes [Renz,Widerspick,Dräger’20,’21].
- 24. Open code - open science MMCS implemented in package volesti of Geomscale Org. github.com/GeomScale/volume_approximation. geomscale.github.io/. Thank you!