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PhD oral defense_Yang

  1. 1. Epidemic Disease Phyllotactic Patterns Gene Regulatory Networks Metabolic Networks Optimization Models and Computational Methods for Systems Biology PhD candidate: Cong Yang Supervisors: Dr. Ching Wai-Ki, Dr. Tsing Nam-Kiu Jan 17th 2012 PhD candidate: Cong Yang Optimization Models and Computational Methods for Systems Bi
  2. 2. Epidemic Disease Phyllotactic Patterns Gene Regulatory Networks Metabolic Networks Systems biology Systems biology treats a biological system as a network, and focuses on all its components and their interactions. Research topics 1 Dynamics and control of the spread of HIV in a system of prisons 2 Dynamics and robustness of phyllotactic patterns 3 Long-run behavior and control of gene regulatory networks 4 Robustness of metabolic networks PhD candidate: Cong Yang Optimization Models and Computational Methods for Systems Bi
  3. 3. Epidemic Disease Phyllotactic Patterns Gene Regulatory Networks Metabolic Networks Dynamics of HIV spread in a system of prisons Dynamics of HIV spread under control policy Optimal control policy Section I: Epidemic Disease 1 Background and main reference 2 Dynamics of HIV spreading in a system of prisons general model of HIV spreading Newton’s method and equilibrium point 3 Dynamics of HIV spreading in a system of prisons under screening and quarantine policy general model of HIV spreading under screening and quarantine existence and stability of equilibrium point numerical examples 4 Optimal screening and quarantine policy PhD candidate: Cong Yang Optimization Models and Computational Methods for Systems Bi
  4. 4. Epidemic Disease Phyllotactic Patterns Gene Regulatory Networks Metabolic Networks Dynamics of HIV spread in a system of prisons Dynamics of HIV spread under control policy Optimal control policy Background and main reference The spread of HIV by both sexual contacts and needle sharing is a serious problem in prisons . [39] J. Gani, S. Yakowitz and M. Blount, The Spread and quarantine of HIV infection in a prison system, SIAM J Appl Math., 57(6)(1997):1510-1530. 1 deterministic model and stochastic analogue for a single prison and a two-prison system focus on the change of HIV infectives along with time, without considering the existence of equilibriums 2 screening and quarantine in a prison and cost-effectiveness analysis quarantine all the HIV infectives by screening PhD candidate: Cong Yang Optimization Models and Computational Methods for Systems Bi
  5. 5. Epidemic Disease Phyllotactic Patterns Gene Regulatory Networks Metabolic Networks Dynamics of HIV spread in a system of prisons Dynamics of HIV spread under control policy Optimal control policy For a single prison without control manner N the number of prisoners y(t) the number of infectives β the infection rate µ the mean proportion of infectives in the outside world n the number of prisoners exchanged with the outside world dy dt = βy(t) N − y(t) new infectives − n N y(t) outflow + nµ inflow [39] J. Gani, S. Yakowitz and M. Blount, The Spread and quarantine of HIV infection in a prison system, SIAM J Appl Math., 57(6)(1997):1510-1530. PhD candidate: Cong Yang Optimization Models and Computational Methods for Systems Bi
  6. 6. Epidemic Disease Phyllotactic Patterns Gene Regulatory Networks Metabolic Networks Dynamics of HIV spread in a system of prisons Dynamics of HIV spread under control policy Optimal control policy For a system of s prisons without control manner Ni , the number of prisoners in Prison i yi (t), the number of infectives in Prisont βi , the infection rate in Prison i µ, the proportion of infectives in the outside world ni , the number of prisoners exchanged between the outside world and Prison i m, the number of prisoners exchanged between any two prisons dyi (t) dt = βi yi (t)(Ni − yi (t)) new infectives − m(s − 1)yi (t) Ni − ni yi (t) Ni outflow + s k=1,k=i yk (t)m Nk + ni µ inflow PhD candidate: Cong Yang Optimization Models and Computational Methods for Systems Bi
  7. 7. Epidemic Disease Phyllotactic Patterns Gene Regulatory Networks Metabolic Networks Dynamics of HIV spread in a system of prisons Dynamics of HIV spread under control policy Optimal control policy Equilibrium point Let (y∗ 1 , y∗ 2 , . . . , y∗ s ) be the non-negative equilibrium point, if N1 = N2 = . . . = Ns and βi < ni N2 i then the equilibrium point is asymptotically stable. Newton’s method A sufficient condition for Newton’s method to be convergent with the initial guess a = 1 2(N1, N2, . . . , Ns)t is s i=1 β2 i s i=1 ni (µ − 1 2 ) + βi N2 i 4 2 2 s i=1 Ni ni 4 ≤ 1 16 . PhD candidate: Cong Yang Optimization Models and Computational Methods for Systems Bi
  8. 8. Epidemic Disease Phyllotactic Patterns Gene Regulatory Networks Metabolic Networks Dynamics of HIV spread in a system of prisons Dynamics of HIV spread under control policy Optimal control policy For a single prison with control manner N the number of prisoners y(t) the number of infected but not detected m(t) the number of detected but not quarantined q(t) the number of quarantined x(t) = N − y(t) − m(t) − q(t) the number of suspected β the infection rate µ the mean proportion of infectives in the outside world n the number of prisoners exchanging with the outside world τ the proportion of prisoners to be screened (0 ≤ τ < 1) κ the proportion of detected to be quarantined (0 ≤ κ ≤ 1) PhD candidate: Cong Yang Optimization Models and Computational Methods for Systems Bi
  9. 9. Epidemic Disease Phyllotactic Patterns Gene Regulatory Networks Metabolic Networks Dynamics of HIV spread in a system of prisons Dynamics of HIV spread under control policy Optimal control policy Dynamics of HIV spreading in a single prison under screening and quarantine    y(t + 1) = (1 − τ)(1 − n N ) (y(t) + βx(t)(y(t) + m(t))) + (1 − τ)nµ m(t + 1) = (1 − κ)(1 − n N )m(t) + (1 − κ)τnµ +(1 − κ)τ(1 − n N )(y(t) + βx(t)(y(t) + m(t))) q(t + 1) = (1 − n N )q(t) + κ(1 − n N )m(t) +κτ(1 − n N )(y(t) + βx(t)(y(t) + m(t))) + τκµn x(t + 1) = N − y(t + 1) − m(t + 1) − q(t + 1). Existence and stability of equilibrium point Given that µ < N 2N+n , The equilibrium point exists. The equilibrium point is asymptotically stable. µ is the mean proportion of infectives in the outside world PhD candidate: Cong Yang Optimization Models and Computational Methods for Systems Bi
  10. 10. Epidemic Disease Phyllotactic Patterns Gene Regulatory Networks Metabolic Networks Dynamics of HIV spread in a system of prisons Dynamics of HIV spread under control policy Optimal control policy Efficiency of the control strategy N = 500, β = 0.0005, n = 50, y(0) = 50, µ = 0.01, τ = 0.1, κ = 0.1. PhD candidate: Cong Yang Optimization Models and Computational Methods for Systems Bi
  11. 11. Epidemic Disease Phyllotactic Patterns Gene Regulatory Networks Metabolic Networks Dynamics of HIV spread in a system of prisons Dynamics of HIV spread under control policy Optimal control policy For a system of s prisons with control manner Ni , the number of prisoners in Prison i yi (t), the number of infected but not detected in Prison i mi (t), the number of detected but not quarantined in Prison i qi (t), the number of quarantined in Prison i xi (t) = Ni − yi (t) − mi (t) − qi (t), the number of suscepted in Prison i βi , the infection rate in Prison i µ, the mean proportion of infectives in the outside world ni , the number of prisoners in Prison i exchanging with the outside world h, the number of prisoners exchange between any two prisons τ the proportion of prisoners to be screened (0 ≤ τ < 1) κ the proportion of detected to be quarantined (0 ≤ κ ≤ 1) PhD candidate: Cong Yang Optimization Models and Computational Methods for Systems Bi
  12. 12. Epidemic Disease Phyllotactic Patterns Gene Regulatory Networks Metabolic Networks Dynamics of HIV spread in a system of prisons Dynamics of HIV spread under control policy Optimal control policy Dynamics of HIV spreading in a system of s prisons under screening and quarantine    yi (t + 1) = (1 − τ)(1 − ni +(s−1)h Ni )(yi (t) + βi xi (t)(yi (t) + mi (t))) +(1 − τ) s j=1,j=i h Nj (yj (t) + βj xj (t)(yj (t) + mj (t))) + (1 − τ)ni µ mi (t + 1) = (1 − κ)(1 − ni +(s−1)h Ni )mi (t) + (1 − κ) s j=1,j=i h Nj mj (t) +(1 − κ)τ(1 − ni +(s−1)h Ni )(yi (t) + βi xi (t)(yi (t) + mi (t))) +(1 − κ)τ s j=1,j=i h Nj (yj (t) + βj xj (t)(yj (t) + mj (t))) + (1 − κ)τni µ qi (t + 1) = (1 − ni +(s−1)h Ni )qi (t) + s j=1,j=i h Nj qj (t) + κ(1 − ni +(s−1)h Ni )mi (t) +κ s j=1,j=i h Nj mj (t)) + κτ(1 − ni +(s−1)h Ni )(yi (t) + βi xi (t)(yi (t) + mi (t))) +κτ s j=1,j=i h Nj (yj (t) + βj xj (t)(yj (t) + mj (t))) + κτni µ i = 1, 2, . . . , s PhD candidate: Cong Yang Optimization Models and Computational Methods for Systems Bi
  13. 13. Epidemic Disease Phyllotactic Patterns Gene Regulatory Networks Metabolic Networks Dynamics of HIV spread in a system of prisons Dynamics of HIV spread under control policy Optimal control policy Existence and stability of equilibrium point 1 Without exchange of prisoners h = 0 If µ < Ni 2Ni +ni , i = 1, 2, . . . , s holds, then Equilibrium solution exists. The equilibrium point is asymptotically stable. 2 With exchange of prisoners h = 0 Given that µ < Ni 2Ni +ni , if h Ni for i = 1, 2, . . . , s holds, then the equilibrium point is asymptotically stable. µ is the mean proportion of infectives in the outside world PhD candidate: Cong Yang Optimization Models and Computational Methods for Systems Bi
  14. 14. Epidemic Disease Phyllotactic Patterns Gene Regulatory Networks Metabolic Networks Dynamics of HIV spread in a system of prisons Dynamics of HIV spread under control policy Optimal control policy A two-prison system Assumption: Screening cost Csi (τ, Ni ) = aτNi and quarantining cost Cqi (κ, Ni ) = bκNi only depend on Ni . Optimization Problem    min C(τ, κ, N) = (aτ + bκ)(N1 + N2) s.t. qi ≤ Qi , i = 1, 2 2 i=1 yi + mi + qi ≤ I 0 ≤ τ, κ ≤ 1. grid search (τ, κ) ∈ {(0.01i, 0.01j) : i, j = 0, 1, . . . , 100}. PhD candidate: Cong Yang Optimization Models and Computational Methods for Systems Bi
  15. 15. Epidemic Disease Phyllotactic Patterns Gene Regulatory Networks Metabolic Networks Generation of a new primordium Interaction among primordia Measurement of pattern uniformity Robustness Section II: Phyllotactic Patterns 1 Background 2 Generation of a new primordium Hofmeister’s hypothesis (MaxMin principle) Atela’s Max-Min model a simplified Max-Min model 3 Interaction among primordia during growing process Ridley’s contact pressure model repulsion pressure model 4 Measurement of pattern uniformity 5 Robustness PhD candidate: Cong Yang Optimization Models and Computational Methods for Systems Bi
  16. 16. Epidemic Disease Phyllotactic Patterns Gene Regulatory Networks Metabolic Networks Generation of a new primordium Interaction among primordia Measurement of pattern uniformity Robustness Phyllotactic patterns (a) sunflower pattern (b) microscope image (c) simulated pattern 1 Where the new primordium appears? 2 How the primordia influence each other during the growing process? PhD candidate: Cong Yang Optimization Models and Computational Methods for Systems Bi
  17. 17. Epidemic Disease Phyllotactic Patterns Gene Regulatory Networks Metabolic Networks Generation of a new primordium Interaction among primordia Measurement of pattern uniformity Robustness Mathematical interpretation The apex is represented by a circle with radius R0. Primordia are generated at the periphery of the apex with periodicity T. ti the birth time of the i-th primordium θi the angle between the straight line joining the origin and the i-th primordium and the positive x axis The distance between the apex center and the primordium at time t after birth is R0 + vts , where v is the growing velocity and s = 0.5. At the birth of the k-th primordium tk (k > i) (R0 cos θ, R0 sin θ) The Euclidean distance between the i-th and k-th primordia is ˜di = (R0 + v √ tk − ti ) cos θi − R0 cos θ 2 + (R0 + v √ tk − ti ) sin θi − R0 sin θ 2 . PhD candidate: Cong Yang Optimization Models and Computational Methods for Systems Bi
  18. 18. Epidemic Disease Phyllotactic Patterns Gene Regulatory Networks Metabolic Networks Generation of a new primordium Interaction among primordia Measurement of pattern uniformity Robustness Hofmeister’s hypothesis (MaxMin principle) The new primordium should be born at the position with the largest minimum distance with all other pre-existing primordia. [49] W.F.B. Hofmeister, Allgemeine Morphologie Der Gew˝achse, Kessinger Publishing, 1868. Mathematical interpretation Atela’s Max-Min model θk = max 0≤θ<2π min i∈{1,2,...,k−1} ˜di [7] P. Atela, C. Gol´e and S. Hotton, A dynamical system for plant pattern formation: a rigorous analysis, J Nonlinear Sci., 12(6)(2002): 641-676. Simplified Max-Min model θk = max 0≤θ<2π min i=k−1,k−2 ˜di cos(θk − θk−1) = (1+τ √ T)− (1+τ √ T)2+4(1+τ √ 2T)(2+τ(4 √ 2T−2 √ T)+τ2T) 4(1+τ √ 2T) , τ = v/R0. PhD candidate: Cong Yang Optimization Models and Computational Methods for Systems Bi
  19. 19. Epidemic Disease Phyllotactic Patterns Gene Regulatory Networks Metabolic Networks Generation of a new primordium Interaction among primordia Measurement of pattern uniformity Robustness Interaction among primordia Ridley’s contact pressure model Contact pressure exists when two primordia are close enough. [93] J.N. Ridley, Computer simulation of contact pressure in capitula, J Theor Biol., 95(1)(1982):1-11. repulsion pressure model Repulsion pressure a d2 ij exists between any two primordia. a is the repulsion weight dij is the Euclidean distance The position of the i-th primordium is influenced by all other primordia (repulsion by all others) the nearest left and nearest right neighbors (repulsion by two) PhD candidate: Cong Yang Optimization Models and Computational Methods for Systems Bi
  20. 20. Epidemic Disease Phyllotactic Patterns Gene Regulatory Networks Metabolic Networks Generation of a new primordium Interaction among primordia Measurement of pattern uniformity Robustness Comparison among mechanisms PhD candidate: Cong Yang Optimization Models and Computational Methods for Systems Bi
  21. 21. Epidemic Disease Phyllotactic Patterns Gene Regulatory Networks Metabolic Networks Generation of a new primordium Interaction among primordia Measurement of pattern uniformity Robustness Measure of pattern uniformity 1 map the packing pattern into a disk with radius 1 ˜Ri = Ri /R1, i = 1, 2, . . . , n 2 uniformity value is defined as rmax dmin rmax the radius of the largest circle that can be placed into the pattern without covering any primordium dmin the minimum distances between any two primordia dmin = mini=j d(i, j) PhD candidate: Cong Yang Optimization Models and Computational Methods for Systems Bi
  22. 22. Epidemic Disease Phyllotactic Patterns Gene Regulatory Networks Metabolic Networks Generation of a new primordium Interaction among primordia Measurement of pattern uniformity Robustness Repulsion by all others PhD candidate: Cong Yang Optimization Models and Computational Methods for Systems Bi
  23. 23. Epidemic Disease Phyllotactic Patterns Gene Regulatory Networks Metabolic Networks Generation of a new primordium Interaction among primordia Measurement of pattern uniformity Robustness Repulsion by all others PhD candidate: Cong Yang Optimization Models and Computational Methods for Systems Bi
  24. 24. Epidemic Disease Phyllotactic Patterns Gene Regulatory Networks Metabolic Networks Generation of a new primordium Interaction among primordia Measurement of pattern uniformity Robustness Repulsion by nearest left and right neighbors PhD candidate: Cong Yang Optimization Models and Computational Methods for Systems Bi
  25. 25. Epidemic Disease Phyllotactic Patterns Gene Regulatory Networks Metabolic Networks Generation of a new primordium Interaction among primordia Measurement of pattern uniformity Robustness Repulsion by nearest left and right neighbors PhD candidate: Cong Yang Optimization Models and Computational Methods for Systems Bi
  26. 26. Epidemic Disease Phyllotactic Patterns Gene Regulatory Networks Metabolic Networks Generation of a new primordium Interaction among primordia Measurement of pattern uniformity Robustness Robustness PhD candidate: Cong Yang Optimization Models and Computational Methods for Systems Bi
  27. 27. Epidemic Disease Phyllotactic Patterns Gene Regulatory Networks Metabolic Networks Distribution and enumeration of attractors in PBN Finite-horizon control with multiple hard-constraints Section III: Gene Regulatory Networks 1 Distribution and enumeration of attractors in PBN Background of the problem Estimation of the number of singleton attractors Algorithms for finding attractors Finding singleton attractors Finding cyclic attractors with a fixed period Computational experiments 2 Finite-horizon control with multiple hard-constraints Background of the problem Problem formation Algorithms for finding optimal control strategies Reserving Place Algorithm Genetic Algorithm Computational experiments PhD candidate: Cong Yang Optimization Models and Computational Methods for Systems Bi
  28. 28. Epidemic Disease Phyllotactic Patterns Gene Regulatory Networks Metabolic Networks Distribution and enumeration of attractors in PBN Finite-horizon control with multiple hard-constraints Background of the problem Consider a PBN with n genes v1, v2, . . . , vn. For gene vi , there are l(i) Boolean functions f (i) ji (1 ≤ ji ≤ l(i)) to be randomly assigned. There are N = n i=1 l(i) possible realization fj = (f (1) j1 , f (2) j2 , . . . , f (n) jn ) for the PBN. Attrators In a PBN, a set of Gene Activity Profiles (GAPs) {v1, v2, . . . , vp} is called an attractor of period p, if Prob(v(t + 1) = vi+1 | v(t) = vi ) = 0 holds for all i = 1, . . . , p. PhD candidate: Cong Yang Optimization Models and Computational Methods for Systems Bi
  29. 29. Epidemic Disease Phyllotactic Patterns Gene Regulatory Networks Metabolic Networks Distribution and enumeration of attractors in PBN Finite-horizon control with multiple hard-constraints Estimation of the number of singleton attractors If the numbers of possible Boolean functions for gene vi are identical (l(i) = L, i = 1, . . . , n), then the number of singleton attractors of the PBN is 2 − 1 2 L−1 n . Finding singleton attractors Initialize m = 1 Procedure PBNAttractor( v(t), m) if m = n + 1 then output v1(t), v2(t), . . . , vn(t) return for b = 0 to 1 do vm(t) := b if vi (t) = f (i) j (v(t))(= vi (t + 1)) holds for any j for some i ≤ m then continue else PBNAttractor(v(t), m + 1) return PhD candidate: Cong Yang Optimization Models and Computational Methods for Systems Bi
  30. 30. Epidemic Disease Phyllotactic Patterns Gene Regulatory Networks Metabolic Networks Distribution and enumeration of attractors in PBN Finite-horizon control with multiple hard-constraints Assumptions and notations The numbers of possible Boolean functions for gene vi are identical, i.e. l(i) = L holds for i = 1, 2, . . . , n. For gene vi , the probabilities for choosing any Boolean function f (i) ji are identical c (i) 1 = c (i) 2 = · · · = c (i) l(i). The maximum indegree of gene vi is bounded by constant K. Computational complexity For gene vi , IN(f (i) j ) denotes the set of input genes for corresponding Boolean function f (i) j . General Case: If IN(f (i) j1 ) = IN(f (i) j2 ) (1 ≤ j1, j2 ≤ l(i)) stands for some gene vi , then the number of recursive calls executed for the first m genes is bounded by {(2 − 0.5L−1 sKL )s }n , where s = m/n. Special Case: If IN(f (i) j1 ) = IN(f (i) j2 ) (1 ≤ j1, j2 ≤ l(i)) holds for any gene vi , then the number of recursive calls executed for the first m genes is bounded by {(2 − 0.5L−1 sK )s }n , where s = m/n. PhD candidate: Cong Yang Optimization Models and Computational Methods for Systems Bi
  31. 31. Epidemic Disease Phyllotactic Patterns Gene Regulatory Networks Metabolic Networks Distribution and enumeration of attractors in PBN Finite-horizon control with multiple hard-constraints Finding attractors with period p p-ancestor(vi , j1, . . . , jp): a set of nodes along paths reaching vi with length p when the realization of the PBN at time t + q (1 ≤ q ≤ p) is given by fjq . PhD candidate: Cong Yang Optimization Models and Computational Methods for Systems Bi
  32. 32. Epidemic Disease Phyllotactic Patterns Gene Regulatory Networks Metabolic Networks Distribution and enumeration of attractors in PBN Finite-horizon control with multiple hard-constraints Assumptions and notations 1 The numbers of possible Boolean functions for gene vi are identical, i.e. l(i) = L holds for i = 1, 2, . . . , n. 2 For gene vi , the probabilities for choosing any Boolean function f (i) ji are identical c (i) 1 = c (i) 2 = · · · = c (i) l(i) . 3 The maximum indegree of gene vi is bounded by constant K. Computational complexity For gene vi , IN(f (i) j ) denotes the set of input genes for corresponding Boolean function f (i) j . General Case: If IN(f (i) j1 ) = IN(f (i) j2 ) (1 ≤ j1, j2 ≤ l(i)) stands for some gene vi , then the number of recursive calls executed for the first m genes is bounded by 2 − 0.5L 1−Kp 1−K −1s L 1−Kp 1−K × K(1−Kp) 1−K s n , where s = m/n. Special Case: If IN(f (i) j1 ) = IN(f (i) j2 ) (1 ≤ j1, j2 ≤ l(i)) holds for any gene vi , then the number of recursive calls executed for the first m genes is bounded by 2 − 0.5L−1s K(1−Kp) 1−K s n , where s = m/n. PhD candidate: Cong Yang Optimization Models and Computational Methods for Systems Bi
  33. 33. Epidemic Disease Phyllotactic Patterns Gene Regulatory Networks Metabolic Networks Distribution and enumeration of attractors in PBN Finite-horizon control with multiple hard-constraints Number of singleton attractors The number of singleton attractors of the PBN is 2 − 1 2 L−1 n . based on 100 randomly generated PBN based on 1000 randomly generated PBN for p = 1, L = 2 PhD candidate: Cong Yang Optimization Models and Computational Methods for Systems Bi
  34. 34. Epidemic Disease Phyllotactic Patterns Gene Regulatory Networks Metabolic Networks Distribution and enumeration of attractors in PBN Finite-horizon control with multiple hard-constraints For WNT5A network For WNT5A network, 10,000 PBNs with 2 Boolean functions and 3 inputs for each node are generated based on the structure of WNT5A network. For comparison supers, we randomly generated 10,000 PBNs with 10 nodes and 2 Boolean functions with 3 inputs for each node. In WNT5A network, IN(f (i) j1 ) = IN(f (i) j2 ) stands. PhD candidate: Cong Yang Optimization Models and Computational Methods for Systems Bi
  35. 35. Epidemic Disease Phyllotactic Patterns Gene Regulatory Networks Metabolic Networks Distribution and enumeration of attractors in PBN Finite-horizon control with multiple hard-constraints Section III: Gene Regulatory Networks 1 Distribution and enumeration of attractors in PBN Background of the problem Estimation of the number of singleton attractors Algorithms for finding attractors Finding singleton attractors Finding cyclic attractors with a fixed period Computational experiments 2 Finite-horizon control with multiple hard-constraints Background of the problem Problem formation Algorithms for finding optimal control strategies Reserving Place Algorithm Genetic Algorithm Computational experiments PhD candidate: Cong Yang Optimization Models and Computational Methods for Systems Bi
  36. 36. Epidemic Disease Phyllotactic Patterns Gene Regulatory Networks Metabolic Networks Distribution and enumeration of attractors in PBN Finite-horizon control with multiple hard-constraints Background of the problem Goal manipulate external control approaches to alter the dynamics of the genetic regulatory network with minimum cost Constraints the number of times each control manner can be applied of the network [20] W.K. Ching, S.Q. Zhang, Y. Jiao, T. Akutsu, N.K. Tsing and A.S. Wong, Optimal control policy for Probabilistic Boolean Networks with hard constraints, IET Syst Biol., 3(2)(2009):90-99. Notations The initial probability distribution is x0 = (v1(0), . . . , vn(0))t . At time j, Control ij ∈ {0, 1, 2} is applied to the system. The control sequence from time 1 to T is σ = i1 i2 . . . iT . Vector xT = (v1(T), . . . , vn(T))t = PiT . . . Pi1 x0 is the state distribution at time T by control strategy σ. Set S contains all the possible states of the PBN. Target states S ⊆ S PhD candidate: Cong Yang Optimization Models and Computational Methods for Systems Bi
  37. 37. Epidemic Disease Phyllotactic Patterns Gene Regulatory Networks Metabolic Networks Distribution and enumeration of attractors in PBN Finite-horizon control with multiple hard-constraints Determine a sequence of actions σ over a finite time horizon T leads the system fall into a set of target states S with probability higher than ¯p, i.e. i∈S [xT ]i ≥ ¯p, minimizes the total control cost T i=1 C(σi ). Problem formation    min σ T i=1 C(σi ) s.t. i∈S [xT ]i ≥ ¯p, 0 ≤ s1 ≤ K1, 0 ≤ s2 ≤ K2. Here si is the number of times that Control i is conducted, and Ki is the maximum number of times that Control i can be applied, i = 1, 2. PhD candidate: Cong Yang Optimization Models and Computational Methods for Systems Bi
  38. 38. Epidemic Disease Phyllotactic Patterns Gene Regulatory Networks Metabolic Networks Distribution and enumeration of attractors in PBN Finite-horizon control with multiple hard-constraints Reserving Place Algorithm 1 generate set U = {σ = i1 i2 . . . iT , ij ∈ {0, 1, 2}, and 0 ≤ si ≤ Ki , i = 1, 2} 1 fix 0 ≤ k ≤ K2 place for Control 2 (represented by 2) in the control string of length T 2 place Control 0 and Control 1 in binary string of length T − k string 11 . . . 1 K1 00 . . . 0 T−K1−k is the largest one after decimalization translate decimal digits from 0 to 2T−k − 1 to binary digits of length T − k while checking the number of times that Control 1 is applied 3 increasing k from 0 to K2 2 find the feasible solution set V = {σ ∈ U : i∈S [xT ]i ≥ ¯p} ⊆ U Computational cost The computation cost of the Reserving Place Algorithm is bounded above by MT22n , where M = K2 j=0 K1 i=0 T! i!j!(T − i − j)! . PhD candidate: Cong Yang Optimization Models and Computational Methods for Systems Bi
  39. 39. Epidemic Disease Phyllotactic Patterns Gene Regulatory Networks Metabolic Networks Distribution and enumeration of attractors in PBN Finite-horizon control with multiple hard-constraints Computational experiment    min σ 10 i=1 C(σi ) s.t. [x]3 + [x]4 ≥ 0.8, 0 ≤ s1 ≤ 5, 0 ≤ s2 ≤ 2. x0 = (0.1, 0.4, 0.3, 0.2)t . The costs for conducting Control 1, Control 2 and Control 3 are 2.5, 4 and 0 respectively. PhD candidate: Cong Yang Optimization Models and Computational Methods for Systems Bi
  40. 40. Epidemic Disease Phyllotactic Patterns Gene Regulatory Networks Metabolic Networks Distribution and enumeration of attractors in PBN Finite-horizon control with multiple hard-constraints Reserving Place Algorithm Comparison among two algorithms PhD candidate: Cong Yang Optimization Models and Computational Methods for Systems Bi
  41. 41. Epidemic Disease Phyllotactic Patterns Gene Regulatory Networks Metabolic Networks Impact degrees in metabolic network with cycles Approximation for impact degree distribution Section IV: Metabolic Networks 1 Impact degrees in metabolic network with cycles Background of the problem Definition of impact degree Algorithm design Impact of single deletion Impact of multiple deletion Computational experiments 2 Approximation for impact degree distribution Background of the problem Branching Process Models The Poisson model Power-law model Parameter extraction Evaluation of approximation PhD candidate: Cong Yang Optimization Models and Computational Methods for Systems Bi
  42. 42. Epidemic Disease Phyllotactic Patterns Gene Regulatory Networks Metabolic Networks Impact degrees in metabolic network with cycles Approximation for impact degree distribution Section IV: Metabolic Networks 1 Impact degrees in metabolic network with cycles Background of the problem Definition of impact degree Algorithm design Impact of single deletion Impact of multiple deletion Computational experiments 2 Approximation for impact degree distribution Background of the problem Branching Process Models The Poisson model Power-law model Parameter extraction Evaluation of approximation PhD candidate: Cong Yang Optimization Models and Computational Methods for Systems Bi
  43. 43. Epidemic Disease Phyllotactic Patterns Gene Regulatory Networks Metabolic Networks Impact degrees in metabolic network with cycles Approximation for impact degree distribution Background A metabolic network is composed of its compounds, linking each other by reactions. One can disable (delete) a reaction by disrupting a gene corresponding to the enzyme which catalyzes the reaction. Impact degree Assume all the reactions and compounds are activated, the impact degree of reaction Ri is the number of reactions (including Ri ) that is inactivated due to the deletion of Ri . [62] D. Jiang, S. Zhou and Y. Chen, Compensatory ability to null mutation in metabolic networks, Biotechnol Bioeng., 103(2)(2009):361-369. PhD candidate: Cong Yang Optimization Models and Computational Methods for Systems Bi
  44. 44. Epidemic Disease Phyllotactic Patterns Gene Regulatory Networks Metabolic Networks Impact degrees in metabolic network with cycles Approximation for impact degree distribution Reactions and compounds For each reaction, there are consumed, produced and directly unrelated compounds. Reaction should be inactivated if any consumed compound or produced compound is inactivated. For each compound, there are consuming, producing and directly unrelated reaction. Compound should be inactivated if all its consuming reactions or all its producing reactions are inactivated. PhD candidate: Cong Yang Optimization Models and Computational Methods for Systems Bi
  45. 45. Epidemic Disease Phyllotactic Patterns Gene Regulatory Networks Metabolic Networks Impact degrees in metabolic network with cycles Approximation for impact degree distribution Simple Algorithm PhD candidate: Cong Yang Optimization Models and Computational Methods for Systems Bi
  46. 46. Epidemic Disease Phyllotactic Patterns Gene Regulatory Networks Metabolic Networks Impact degrees in metabolic network with cycles Approximation for impact degree distribution Consider the impact of deleting reaction Ri only related reactions all reactions inactivated inactivated compounds all compounds inactivated related compounds all substrates and products of all related reactions remained compounds the related compounds but not inactivated Deletion of reaction pair (Rh, Rg ) Overlapped compounds compounds that are remained compounds for both reaction Rg and reaction Rh. PhD candidate: Cong Yang Optimization Models and Computational Methods for Systems Bi
  47. 47. Epidemic Disease Phyllotactic Patterns Gene Regulatory Networks Metabolic Networks Impact degrees in metabolic network with cycles Approximation for impact degree distribution Improved algorithm PhD candidate: Cong Yang Optimization Models and Computational Methods for Systems Bi
  48. 48. Epidemic Disease Phyllotactic Patterns Gene Regulatory Networks Metabolic Networks Impact degrees in metabolic network with cycles Approximation for impact degree distribution Medium-scale E. coli network from KEGG 253 reactions (150 are reversible) and 261 compounds average impact degree of single deletion: 1.9331. average impact degree of double deletion: 3.8461. 99.73% simplified case Simple Algorithm: 3427.7 sec, Improved Algorithm: 88.9 sec PhD candidate: Cong Yang Optimization Models and Computational Methods for Systems Bi
  49. 49. Epidemic Disease Phyllotactic Patterns Gene Regulatory Networks Metabolic Networks Impact degrees in metabolic network with cycles Approximation for impact degree distribution Whole E. coli network single deletion double deletion triple deletion PhD candidate: Cong Yang Optimization Models and Computational Methods for Systems Bi
  50. 50. Epidemic Disease Phyllotactic Patterns Gene Regulatory Networks Metabolic Networks Impact degrees in metabolic network with cycles Approximation for impact degree distribution Section IV: Metabolic Networks 1 Impact degrees in metabolic network with cycles Background of the problem Definition of impact degree Algorithm design Impact of single deletion Impact of multiple deletion Computational experiments 2 Approximation for impact degree distribution Background of the problem Branching Process Models The Poisson model Power-law model Parameter extraction Evaluation of approximation PhD candidate: Cong Yang Optimization Models and Computational Methods for Systems Bi
  51. 51. Epidemic Disease Phyllotactic Patterns Gene Regulatory Networks Metabolic Networks Impact degrees in metabolic network with cycles Approximation for impact degree distribution Background Estimation of impact degree distribution of single reaction deletion. The impact spread out in a cascade manner, it spreads through downstream reactions whose substrates are synthesized via unique metabolic reactions. Branching process: each progenitor generates offsprings independently but according to the same probability distribution. PhD candidate: Cong Yang Optimization Models and Computational Methods for Systems Bi
  52. 52. Epidemic Disease Phyllotactic Patterns Gene Regulatory Networks Metabolic Networks Impact degrees in metabolic network with cycles Approximation for impact degree distribution metabolic network corresponding reaction network Offsprings Assume the network is in tree structure. The number of offsprings of reaction Ri di = kout i (if kin i = 1) 0 (otherwise) , where kout i and kin i are the outdegree and indegree of reaction Ri respectively. PhD candidate: Cong Yang Optimization Models and Computational Methods for Systems Bi
  53. 53. Epidemic Disease Phyllotactic Patterns Gene Regulatory Networks Metabolic Networks Impact degrees in metabolic network with cycles Approximation for impact degree distribution The Poisson Model 1 The number of offsprings d follows Poisson distribution, P(d) = µd e−µ /d!, where µ is the mean of the offspring numbers. 2 The total number of offsprings (i.e., impact degree) r follows Borel distribution: P(r) = (µr)r−1 e−µr r! . By Stirling formula, P(r) ∝ r−3/2 e−r(ln µ−µ+1) . PhD candidate: Cong Yang Optimization Models and Computational Methods for Systems Bi
  54. 54. Epidemic Disease Phyllotactic Patterns Gene Regulatory Networks Metabolic Networks Impact degrees in metabolic network with cycles Approximation for impact degree distribution Power-law Model 1 The number of offsprings d follows Power-law distribution, P(d) = C /dγ+1 , where C = 1/ζ(γ + 1). 2 If 1 < γ < 2 holds, the total number of offsprings (i.e., impact degree) r follows P(r) = µ νr1+1/γ ϕ (1 − µ)r − µ − 1 νr1/γ , where ϕ(x) = ∞ 0 exp uγ cos πγ 2 cos uγ sin πγ 2 + ux du, µ is the mean of the number of offsprings, ν = µ[γΓ(−γ)]1/γ, here Γ(x) is the Gamma function. If µ = 1, P(r) ∝ 1/r1+1/γ . If γ > 2, P(r) is similar to that in Poisson model. PhD candidate: Cong Yang Optimization Models and Computational Methods for Systems Bi
  55. 55. Epidemic Disease Phyllotactic Patterns Gene Regulatory Networks Metabolic Networks Impact degrees in metabolic network with cycles Approximation for impact degree distribution Parameter extraction Poisson Model The mean of the number of offsprings for each reaction node µ can be estimated by µ = 1 N N i=1 di , where N is the total number of reaction nodes. Power-law Model The exponent γ can be estimated by γ = |N ∗ |   i∈N∗ ln di dmin   −1 , where N∗ is the set of reaction nodes with di > 0, and |N∗| is the total number of such reaction nodes. dmin is the minimum of di in the set of N∗. PhD candidate: Cong Yang Optimization Models and Computational Methods for Systems Bi
  56. 56. Epidemic Disease Phyllotactic Patterns Gene Regulatory Networks Metabolic Networks Impact degrees in metabolic network with cycles Approximation for impact degree distribution Offspring distribution (A) Escherichia coli, (B) Bacillus subtilis, (C) Saccharomyces cerevisiae and (D) Homo sapiens PhD candidate: Cong Yang Optimization Models and Computational Methods for Systems Bi
  57. 57. Epidemic Disease Phyllotactic Patterns Gene Regulatory Networks Metabolic Networks Impact degrees in metabolic network with cycles Approximation for impact degree distribution Impact degree distribution (A) Escherichia coli, (B) Bacillus subtilis, (C) Saccharomyces cerevisiae and (D) Homo sapiens PhD candidate: Cong Yang Optimization Models and Computational Methods for Systems Bi
  58. 58. Epidemic Disease Phyllotactic Patterns Gene Regulatory Networks Metabolic Networks Impact degrees in metabolic network with cycles Approximation for impact degree distribution Acknowledgment 1 Department of Mathematics, The University of Hong Kong Dr. Tuen-Wai Ng Ho-Yin Leung Dr. Allen H. Tai 2 Bioinformatics Center, Kyoto University Dr. Tatsuya Akutsu Dr. Takeyuki Tamura Dr. Morihiro Hayashida 3 Japan Science and Technology Agency Dr. Kazuhiro Takemoto 4 Centre for Computational Biology, Mines ParisTech Dr. Jean-Philippe Ver 5 Faculty of Mathematical Sciences, Fudan University Dr. Shu-Qin Zhang 6 School of Mathematical Sciences, Xiamen University Dr. Zheng-Jian Bai PhD candidate: Cong Yang Optimization Models and Computational Methods for Systems Bi
  59. 59. Epidemic Disease Phyllotactic Patterns Gene Regulatory Networks Metabolic Networks Impact degrees in metabolic network with cycles Approximation for impact degree distribution Dynamics of HIV spreading in a single prison under screening and quarantine N the number of prisoners y(t) the number of infected but not detected m(t) the number of detected but not quarantined q(t) the number of quarantined x(t) = N − y(t) − m(t) − q(t) the number of suspected β the infection rate µ the mean proportion of infectives in the outside world n the number of prisoners exchanging with the outside world τ the proportion of prisoners to be screened (0 ≤ τ < 1) κ the proportion of detected to be quarantined (0 ≤ κ ≤ 1)    y(t + 1) = (1 − τ)(1 − n N ) (y(t) + βx(t)(y(t) + m(t))) + (1 − τ)nµ m(t + 1) = (1 − κ)(1 − n N )m(t) + (1 − κ)τnµ +(1 − κ)τ(1 − n N )(y(t) + βx(t)(y(t) + m(t))) q(t + 1) = (1 − n N )q(t) + κ(1 − n N )m(t) +κτ(1 − n N )(y(t) + βx(t)(y(t) + m(t))) + τκµn x(t + 1) = N − y(t + 1) − m(t + 1) − q(t + 1). PhD candidate: Cong Yang Optimization Models and Computational Methods for Systems Bi
  60. 60. Epidemic Disease Phyllotactic Patterns Gene Regulatory Networks Metabolic Networks Impact degrees in metabolic network with cycles Approximation for impact degree distribution After exchange with the outside world infected but not detected prisoners (1 − n N )(y(t) + βx(t)(y(t) + m(t))) + nµ detected but not quarantined prisoners (1 − n N )m(t) quarantined prisoners (1 − n N )q(t) After screening of τ of prisoners undetected and unquarantined infected but not detected (1 − τ)(1 − n N )(y(t) + βx(t)(y(t) + m(t))) + (1 − τ)nµ detected but not quarantined (1 − n N )m(t) + τ((1 − n N )(y(t) + βx(t)(y(t) + m(t))) + nµ) quarantined prisoners (1 − n N )q(t) After quarantine of κ of detected prisoners infected but not detected y(t + 1) (1 − τ)(1 − n N )(y(t) + βx(t)(y(t) + m(t))) + (1 − τ)nµ detected but not quarantined m(t + 1) (1 − κ)(1 − n N )m(t) + (1 − κ)τ(1 − n N )(y(t) + βx(t)(y(t) + m(t))) + (1 − κ)τnµ quarantined prisoners q(t + 1) (1 − n N )q(t) + κ(1 − n N )m(t) + κτ(1 − n N )(y(t) + βx(t)(y(t) + m(t))) + τκµn suspected prisoners x(t) N − y(t) − m(t) − q(t) PhD candidate: Cong Yang Optimization Models and Computational Methods for Systems Bi
  61. 61. Epidemic Disease Phyllotactic Patterns Gene Regulatory Networks Metabolic Networks Impact degrees in metabolic network with cycles Approximation for impact degree distribution Newton’s Method For solving F(x) = 0, Xn+1 = Xn − [F (xn)]−1 F(xn). Convergence of Newton’s Method Kantorovich Theorem Let a be a point in RK , U be an open neighborhood of a in RK and F : U → RK be a differential mapping with its derivative [DF(a)] being invertible. Define h = −[DF(a)]−1 F(a), a1 = a + h and U0 = B|h|(a1). If U0 ⊂ U and the derivative [DF(x)] satisfies the Lipschitz condition ||DF(u1) − DF(u2)|| ≤ M|u1 − u2| for all points u1 and u2 and if the inequality |F(a)||DF(a)|−2 M ≤ 1 2 is satisfied, then the equation F(x) = 0 has a unique solution in U0 and Newton’s method converges with an initial guess a. PhD candidate: Cong Yang Optimization Models and Computational Methods for Systems Bi
  62. 62. Epidemic Disease Phyllotactic Patterns Gene Regulatory Networks Metabolic Networks Impact degrees in metabolic network with cycles Approximation for impact degree distribution Stablity Given that xe is the equilibrium point of equation system x = Ax, Lyapunov Stable For ∀ > 0, ∃δ = δ( ) > 0 s.t. if x(0) − xe < δ, then x(t) − xe < for ∀t ≥ 0. Asymptotically Stable Lyapunov stable and if ∃δ > 0 s.t. if x(0) − xe < δ, then lim t→∞ x(t) − xe = 0. Exponentially Stable Asymptotically stable and if ∃α, β, δ > 0 s.t. if x(0) − xe < δ, then x(t) − xe ≤ α x(0) − xe e−βt , t ≥ 0. Asymptotically stable The solution for X = AX is asymptotically stable as t → ∞, if and only if for all the eigenvalues λ of A, Re(λ) < 0. Gershgorin disc Theorem For matrix A = (aij )n×n, closed disc D(aii , Ri ) centered at aii with radius Ri is called Gershgorin disc, where Ri = j=i | aij | is the sum of the absolute values of the non-diagonal entries in the ith row. Every eigenvalue of matrix A lies within at least one of the Gershgorin discs D(aii , Ri ). PhD candidate: Cong Yang Optimization Models and Computational Methods for Systems Bi
  63. 63. Epidemic Disease Phyllotactic Patterns Gene Regulatory Networks Metabolic Networks Impact degrees in metabolic network with cycles Approximation for impact degree distribution Phyllotactic patterns under different s PhD candidate: Cong Yang Optimization Models and Computational Methods for Systems Bi
  64. 64. Epidemic Disease Phyllotactic Patterns Gene Regulatory Networks Metabolic Networks Impact degrees in metabolic network with cycles Approximation for impact degree distribution Repulsion by all others under various repulsion weight a PhD candidate: Cong Yang Optimization Models and Computational Methods for Systems Bi
  65. 65. Epidemic Disease Phyllotactic Patterns Gene Regulatory Networks Metabolic Networks Impact degrees in metabolic network with cycles Approximation for impact degree distribution Repulsion by nearest left and right under various repulsion weight a PhD candidate: Cong Yang Optimization Models and Computational Methods for Systems Bi
  66. 66. Epidemic Disease Phyllotactic Patterns Gene Regulatory Networks Metabolic Networks Impact degrees in metabolic network with cycles Approximation for impact degree distribution Estimation of the number of singleton attractors Assume that the numbers of possible Boolean functions for gene vi are identical, i.e., l(i) = L holds for all i. The expected number of singleton attractors for a PBN with n genes is 2n × 1 − 1 2 L n = 2 − 1 2 L−1 n . If fj (v) = v holds for some realization fj , then GAP v is a singleton attractor. For each gene vi , for any PBN realization fj , the probability that f (i) j (v(t)) = a, a ∈ {0, 1} is (1 2)L. Thus Prob(v(t + 1) = u | v(t) = u) = {1 − (1 2)L}n, and the expected number of singleton attractors is 2n × 1 − 1 2 L n = 2 − 1 2 L−1 n . PhD candidate: Cong Yang Optimization Models and Computational Methods for Systems Bi
  67. 67. Epidemic Disease Phyllotactic Patterns Gene Regulatory Networks Metabolic Networks Impact degrees in metabolic network with cycles Approximation for impact degree distribution Branch and bound algorithm It consists of a systematic enumeration of all candidate solutions, where large subsets of fruitless candidates are discarded, by using upper and lower estimated bounds of the quantity being optimized. Finding singleton attractors Initialize m = 1 Procedure PBNAttractor( v(t), m) if m = n + 1 then output v1(t), v2(t), . . . , vn(t) return for b = 0 to 1 do vm(t) := b if vi (t) = f (i) j (v(t))(= vi (t + 1)) holds for any j for some i ≤ m then continue else PBNAttractor(v(t), m + 1) return PhD candidate: Cong Yang Optimization Models and Computational Methods for Systems Bi
  68. 68. Epidemic Disease Phyllotactic Patterns Gene Regulatory Networks Metabolic Networks Impact degrees in metabolic network with cycles Approximation for impact degree distribution Assumptions and notations The numbers of possible Boolean functions for gene vi are identical, i.e. l(i) = L holds for i = 1, 2, . . . , n. For gene vi , the probabilities for choosing any Boolean function f (i) ji are identical c (i) 1 = c (i) 2 = · · · = c (i) l(i) . The maximum indegree of gene vi is bounded by constant K. For gene vi , IN(f (i) j ) denotes the set of input genes for corresponding Boolean function f (i) j . Singleton attractor IN(f (i) j1 ) = IN(f (i) j2 ) The number of recursive calls executed for the first m genes is bounded by {2 − 0.5L−1 m n KL }m . The probability that for some gene some vi , i ≤ m, vi (t) = f (i) j (v(t))(= vi (t + 1)) holds for any Boolean function f (i) j , 1 ≤ j ≤ l(i) = L is L j=1    0.5 × C |IN(f (i) j )| m C |IN(f (i) j )| n    ≈ L j=1 0.5 × m n |IN(f (i) j )| ≥ 0.5 L m n KL . The probability that the algorithm examines the (m + 1)-th gene is smaller or equal to 1 − 0.5L m n KL m . Thus the number of recursive calls executed for the first m genes is bounded by 2 m × 1 − 0.5 L m n KL m = 2 − 0.5 L−1 m n KL m . PhD candidate: Cong Yang Optimization Models and Computational Methods for Systems Bi
  69. 69. Epidemic Disease Phyllotactic Patterns Gene Regulatory Networks Metabolic Networks Impact degrees in metabolic network with cycles Approximation for impact degree distribution Assumptions and notations The numbers of possible Boolean functions for gene vi are identical, i.e. l(i) = L holds for i = 1, 2, . . . , n. For gene vi , the probabilities for choosing any Boolean function f (i) ji are identical c (i) 1 = c (i) 2 = · · · = c (i) l(i) . The maximum indegree of gene vi is bounded by constant K. For gene vi , IN(f (i) j ) denotes the set of input genes for corresponding Boolean function f (i) j . Singleton attractor IN(f (i) j1 ) = IN(f (i) j2 ) The number of recursive calls executed for the first m genes is bounded by {2 − 0.5L−1 m n K }m . The probability that vi (t) = vi (t + 1) holds for some i is 0.5 L × C |IN(f (i) 1 )| m C |IN(f (i) 1 )| n ≈ 0.5 L m n |IN(f (i) 1 )| ≥ 0.5 L m n K . The probability that the algorithm examines the (m + 1)-th gene is no more than 1 − 0.5L m n K m . The number of recursive calls executed for the first m genes is thus bounded by 2 m × 1 − 0.5 L m n K m = 2 − 0.5 L−1 m n K m . PhD candidate: Cong Yang Optimization Models and Computational Methods for Systems Bi
  70. 70. Epidemic Disease Phyllotactic Patterns Gene Regulatory Networks Metabolic Networks Impact degrees in metabolic network with cycles Approximation for impact degree distribution Assumptions and notations The numbers of possible Boolean functions for gene vi are identical, i.e. l(i) = L holds for i = 1, 2, . . . , n. For gene vi , the probabilities for choosing any Boolean function f (i) ji are identical c (i) 1 = c (i) 2 = · · · = c (i) l(i) . The maximum indegree of gene vi is bounded by constant K. For gene vi , IN(f (i) j ) denotes the set of input genes for corresponding Boolean function f (i) j . attractor of period p IN(f (i) j1 ) = IN(f (i) j2 ) The number of recursive calls executed for the first m genes is bounded by      2 − 0.5 L 1−Kp 1−K −1 s L 1−Kp 1−K × K(1−Kp) 1−K    s    n , where s = m/n. At the corresponding time, one out of L Boolean functions is chosen for each node of p-ancestor. There are at most Kp input nodes. There are at most L p−1 q=0 Kq = L(1−Kp)/(1−K) combinations of p realizations for each node vi . The probability that vi (t) = vi (t + p) holds for some i is approximately at least    0.5 × C p q=1 Kq m C p q=1 Kq n    L(1−Kp)/(1−K) ≈ 0.5 × m n p q=1 Kq L(1−Kp)/(1−K) . PhD candidate: Cong Yang Optimization Models and Computational Methods for Systems Bi
  71. 71. Epidemic Disease Phyllotactic Patterns Gene Regulatory Networks Metabolic Networks Impact degrees in metabolic network with cycles Approximation for impact degree distribution Assumptions and notations The numbers of possible Boolean functions for gene vi are identical, i.e. l(i) = L holds for i = 1, 2, . . . , n. For gene vi , the probabilities for choosing any Boolean function f (i) ji are identical c (i) 1 = c (i) 2 = · · · = c (i) l(i) . The maximum indegree of gene vi is bounded by constant K. For gene vi , IN(f (i) j ) denotes the set of input genes for corresponding Boolean function f (i) j . attractor of period p IN(f (i) j1 ) = IN(f (i) j2 ) The number of recursive calls executed for the first m genes is bounded by 2 − 0.5 L−1 s K(1−Kp) 1−K s n , where s = m/n. The probability that vi (t) = vi (t + p) holds equals to 0.5 L × C p q=1 Kq m C p q=1 Kq n ≈ 0.5 L m n p q=1 Kq . PhD candidate: Cong Yang Optimization Models and Computational Methods for Systems Bi
  72. 72. Epidemic Disease Phyllotactic Patterns Gene Regulatory Networks Metabolic Networks Impact degrees in metabolic network with cycles Approximation for impact degree distribution Reserving Place Algorithm The computation cost of the Reserving Place Algorithm is bounded above by MT22n, where M = K2 j=0 K1 i=0 T! i!j!(T − i − j)! . The matrix-vector multiplication PiT . . . Pi1 x0 occupies the main computational time. If one searches optimal solutions in the set W = {σ = i1 i2 . . . iT : ij ∈ {0, 1, 2}}, then one needs to conduct T3T 22n basic operations, where n is the number of genes in the PBN, where T is the time length. For Reserving Place algorithm, one only needs to find the optimal control strategies in the set V = {σ ∈ U : i∈S [xT ]i ≥ ¯p}. The number of basic operation to be conducted equals to T22nn(V ), where n(V ) is the size of set V . Note that V ⊆ U, the computational cost is bounded above by T22nn(U). For the Reserving Place algorithm, the number of basic operations is upper bounded by MT22n, where M = n(U) = K2 j=0  C j T K1 i=0 C i T−j   = K2 j=0 K1 i=0 T! i!j!(T − i − j)! . PhD candidate: Cong Yang Optimization Models and Computational Methods for Systems Bi
  73. 73. Epidemic Disease Phyllotactic Patterns Gene Regulatory Networks Metabolic Networks Impact degrees in metabolic network with cycles Approximation for impact degree distribution Metabolic network Mechanisms reaction R = (C1 ∧ C2) ∧ (C3 ∧ C4) compound C = (R12 ∨ R2) ∧ (R11 ∨ R3 ∨ R4) = (R1 ∨ R2) ∧ (R1 ∨ R3 ∨ R4) PhD candidate: Cong Yang Optimization Models and Computational Methods for Systems Bi

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