### Solving x’=?

• 2. Context/Motivation (There are different concepts of solving a differential equation)
• 3. Classical Differential Equations Isaac Newton 1643-1727 mi d2 qi dt2 = G X j6=i mjmi(qj qi) kqj qik3 Newton’s Law of Gravitation d2 q1 dt2 = Gm2(q2 q1) kq2 q1k3 d2 q2 dt2 = Gm1(q1 q2) kq1 q2k3 2-body problem Kepler’s three laws of planetary motion Johannes Kepler 1571-1630
• 4. Still Useful Today Restricted 3 Body Problem W.S. Koon, M. W. Lo, J. E. Marsden, S. D. Ross, Heteroclinic connections between periodic orbits and resonance transitions in celestial mechanics, Chaos, 2000 Motivation: “the design of trajectories for space missions such as the Genesis Discovery Mission.” d2 x dt2 2 dy dt = ⌦x d2 y dt2 + 2 dx dt = ⌦y ⌦(x, y) = x2 + y2 2 + 1 µ r1 + µ2 r2 + µ(1 µ) 2 Genesis Discovery Mission Equations involve explicit analytic expressions.
• 5. Worth Noting: We can compute orbits that exhibit fascinating complexity. How does this help our understanding? u(x, y, z, t) is velocity ﬁeld p(x, y, z, t) is pressure ﬁeld T(x, y, z, t) is temperature ﬁeld Pr 1 ✓ @u @t + u · ru ◆ = rp + r2 u + RaTˆz @T @t + u · rT = r2 T r · u = 0 Boussinesq Equations Mark Paul, VA Tech
• 6. The 3-body Problem ≈1890 Jules Henri Poincare 1854-1912 Chaotic dynamics exists. Understanding the solution of a single initial value problem is not sufﬁcient. S ⇢ Rn is an invariant set if '(t, S) = S for all times t. ': R ⇥ Rn ! Rn (t, x) 7! '(t, x) Flow: initial condition time value of solution at time t Consider all solutions: Map: f : Rn ! Rn x 7! f(x) := '(⌧, x) ⌧ > 0 is a ﬁxed time. Examples: equilibria, periodic orbits, connecting orbits, strange attractors
• 8. Estimated number of malaria cases in 2010: between 219 and 550 million Estimated number of deaths due to malaria in 2010: 600,000 to 1,240,000 Malaria may have killed half of all the people that ever lived. And more people are now infected than at any point in history. There are up to half a billion cases every year, and about 2 million deaths - half of those are children in sub-Saharan Africa. J. Whitﬁeld, Nature, 2002 Resistance is now common against all classes of antimalarial drugs apart from artemisinins. … Malaria strains found in ﬁve countries in the Greater Mekong Subregion are resistant to combination therapies that include artemisinins, and may therefore be untreatable. World Health Organization Malaria is of great public health concern, and seems likely to be the vector-borne disease most sensitive to long-term climate change. World Health Organization A Current Problem: Malaria
• 9. Malaria: P. falciparum 48 hour cycle 1-2 minutes All genes (5409) 1.5$0.0$ &1.5$Standard$devia0ons$from$ mean$expression$(z&score)$High Low 0$ 10$20$ 30$40$ 0me$in#vitro#(hours)$ 50$60$ periodic$genes$(43)\$ 10 20 30 40 50 600 Task: Characterize the dynamics with the goal of affecting the dynamics with drugs. A proposed network A differential equation dx dt = f(x, ) is proba- bly a reasonable model for the dynamics, but I do not have an analytic description of f or estimates of the parameters . Malaria is • Sequenced • Poorly annotated
• 10. RB-E2F pathwayCancer Poorly quantiﬁed: biochemistry, e.g. reaction rates, binding energies, etc., not known What is (are) appropriate model(s) for dynamics? This is a dynamic process: timing and sequencing of events is essential Yao, et. al., MSB, 2011 Deregulation of the RB–E2F pathway is implicated in most, if not all, human cancers.
• 11. Biological Model Biological Data/Phenotype Hill functions: 1+4 parameters˙x = f(x, ) x 2 RN , 2 RM Physics/Math Model Yao et.al. follow a traditional approach Yao, et. al., MSB, 2011 Remark: Typical model considered by Yao et.al. has ≈ 30 parameters. Strategy: Choose 20,000 random parameter values and evaluate. Quality of model = QM = # parameters with bistability 20,000 A worry: 230 = 1, 073, 741, 824
• 13. The Lac Operon Ozbudak et al. Nature 2004 Network Model 1 ⌧y ˙y = ↵ RT RT + R(x) y 1 ⌧x ˙x = y x R(x) = RT 1 + ⇣ x x0 ⌘n ODE Model Data ODES are great modeling tools, but should be handled with care. parameter values ↵ = 84.4 1 + (G/8.1)1.2 + 16.1 = . . .
• 14. Classical QualitaIve RepresentaIon of Dynamics Dynamic Signature (Morse Graph) Not Precise Accurate Rigorous Precise Not Accurate Not Rigorous What does it mean to solve an ODE? Conley-Morse Chain Complex model“truth” parameter
• 15. Biological Model Biological Data/Phenotype ˙x = f(x, ) x 2 RN , 2 RM Physics/Math Model Traditional approach Part I Part II Part III Finite Computational Model Order theory Algebraic topologyThis talk:
• 16. Order Theory and Dynamics ; {a} {b} {ac} {ab} {bf} {abc} {abd} {abe} {abf} {abcd} {abde} {abcf} {abef} {abcde} {abcdf} {abcef} {abdef}{abdeg} {abcdeg} {abcdef} {abdefg} {abdefh} {abcdefg} {abcdefh} {abcdefh}
• 18. Let (L, ^, _, 0, 1) denote a ﬁnite bounded distributive lattice. Birkhoff’s Theorem Birkhoff’s Theorem: O(J_ (L)) ⇠= L J_ (O(P)) ⇠= P Let (P, <) denote a partially ordered set (poset). c b’b a P The lattice of down sets of (P, <) is O(P) := {U ⇢ P | if x 2 U and y < x then y 2 U} . {a, b, b0 , c} {a, b, b0 } {a, b0 }{a, b} {a} ; O(P) The poset of join irreducible elements of L is J_ (L) := {x 2 L | if x = a _ b, then a = x or b = x} J_ (O(P))
• 19. Let X be a compact metric space. phase space TopologyDynamics Use Birkhoff to deﬁne poset (P := J_ (A), <) G(L) denoted atoms of L “smallest” elements of L For each p 2 P deﬁne a Morse tile M(p) := cl(A pred(A)) Declare a bounded sublattice A ⇢ L to be a lattice of attracting blocks Space of all approximations Reg(X) denotes the lattice of regular closed subsets of X. L is a ﬁnite bounded atomic sublattice of Reg(X) The chosen approximation scheme
• 20. Example Morse tiles M(p) Let F0 (x) = f(x). -4 40 Atoms of lattice: G(L) = {[n, n + 1] | n = 4, . . . , 3} Phase space: X = [ 4, 4] ⇢ R P 1 2 3 Birkhoff How does this relate to a differential equation dx dt = f(x)? -4 40 F F (bistability) A Lattice of attracting blocks: A = {[ 3, 1], [1, 3], [ 3, 1] [ [1, 3], [ 4, 4]} Attracting blocks are regions of phase space that are forward invariant with time. F
• 21. Biological Model Biological Data/Phenotype ˙x = f(x, ) x 2 RN , 2 RM Physics/Math Model Traditional approach Finite Computational Model Part I Part II Part III Order theory Algebraic topology Dynamic Structures Generated from Regulatory Networks
• 22. p1 p0 p2 p3 Vertices: States Edges: Dynamics Simple decomposition of Dynamics: Recurrent Nonrecurrent (gradient-like) Linear time Algorithm! Morse Graph of state transition graph State Transition Graph F : X !! X An essential computational tool
• 23. p1 p0 p2 p3 P O S E T Morse Graph of F : X !! X Join Irreducible J_ (A) Birkhoff’s Theorem implies that the Morse graph and the lattice of Attractors are equivalent. What is observable? A X is an attractor if F(A) = A p1 p0 p1, p0 p2, p1, p0 p3, p2, p1, p0 Lower Sets O(M) ; Lattice of Attractors of F : X !! X _ = [ ^ = maximal attractor in Com putable Observable
• 24. Biological Model Assume xi decays. dxi dt = ixi dxi dt = ixi + ⇤i(x)dxi dt = ixi + ⇤i(xj) How do I want to interpret this information? What differential equation do I want to use? Proposed model: dx2 dt x1 ✓2,1 u2,1 l2,1 x1 represses the production of x2. 1 2 x1 activates the production of x2. 1 2 Parameters 1/node 3/edge For x1 < ✓2,1 we ask about sign ( 2x2 + u2,1). For x1 > ✓2,1 we ask about sign ( 2x2 + l2,1). xi denotes amount of species i. j,i(xi) = ( uj,i if xi < ✓j,i j,i if xi > ✓j,i Focus on sign of ixi + i,j(xj) ixi + + i,j(xj)
• 25. 12 ✓2,1 ✓1,2 x1 x2 Phase space: X = (0, 1)2 If 1✓2,1 + 1,2(x2) > 0 If 1✓2,1 + 1,2(x2) < 0 Example (The Toggle Switch) Parameter space is a subset of (0, 1)8 Fix z a regular parameter value. z is a regular parameter value if 0 < i 0 < i,j < ui,j, 0 < ✓i,k 6= ✓j,k, and 0 6= i✓j,i + ⇤i(x)
• 26. ✓2,1 ✓1,2 x1 x2 Need to Construct State Transition Graph Fz : X !! X Example (The Toggle Switch) 12 Fix z a regular parameter value. Vertices X corresponds to all rectangular domains and co-dimension 1 faces deﬁned by thresholds ✓. Faces pointing in map to their domain. Domains map to their faces pointing out. Edges If no outpointing faces domain maps to itself.
• 27. 12The Toggle Switch ✓2,1 ✓1,2 x1 x2 Assume: l1,2 < 1✓2,1 < u1,2 2✓1,2 < l2,1 Morse Graph FP{0,1} Fix z a regular parameter value. Constructing state transition graph Fz : X !! X Check signs of i✓j,i + i,j(xj)
• 28. DSGRN Database from Genetic Toggle Switch 12 Input: Regulatory Network Output: DSGRN database Parameter graph provides explicit partition of entire 8-D parameter space. We can query this database for local or global dynamics. Parameter graph is a product graph over each node. (7) FP(1,1) 1✓2,1 < l1,2 < u1,2 2✓1,2 < l2,1 < u2,1 (8) FP(1,0) 1✓2,1 < l1,2 < u1,2 l2,1 < 2✓1,2 < u2,1 (9) FP(1,0) 1✓2,1 < l1,2 < u1,2 u2,1 < u2,1 < 2✓1,2 (4) FP(0,1) l1,2 < 1✓2,1 < u1,2 2✓1,2 < l2,1 < u2,1 (5) FP(0,1) FP(1,0) l1,2 < 1✓2,1 < u1,2 l2,1 < 2✓1,2 < u2,1 (6) FP(1,0) l1,2 < 1✓2,1 < u1,2 l2,1 < u2,1 < 2✓1,2 (1) FP(0,1) l1,2 < u1,2 < 1✓2,1 2✓1,2 < l2,1 < u2,1 (2) FP(0,1) l1,2 < u1,2 < 1✓2,1 l2,1 < 2✓1,2 < u2,1 (3) FP(0,0) l1,2 < u1,2 < 1✓2,1 u2,1 < u2,1 < 2✓1,2
• 29. Why is the Toggle Switch a Switch? x1 x2 ✓2,1 ✓1,2 (0,1) (1,0) 12 FP(0,1) FP(1,0) ✓1,2 x1 switch/hysteresis Paths deﬁned by varying ✓1,2 (7) FP(1,1) 1✓2,1 < l1,2 < u1,2 2✓1,2 < l2,1 < u2,1 (8) FP(1,0) 1✓2,1 < l1,2 < u1,2 l2,1 < 2✓1,2 < u2,1 (9) FP(1,0) 1✓2,1 < l1,2 < u1,2 u2,1 < u2,1 < 2✓1,2 (4) FP(0,1) l1,2 < 1✓2,1 < u1,2 2✓1,2 < l2,1 < u2,1 (5) FP(0,1) FP(1,0) l1,2 < 1✓2,1 < u1,2 l2,1 < 2✓1,2 < u2,1 (6) FP(1,0) l1,2 < 1✓2,1 < u1,2 l2,1 < u2,1 < 2✓1,2 (1) FP(0,1) l1,2 < u1,2 < 1✓2,1 2✓1,2 < l2,1 < u2,1 (2) FP(0,1) l1,2 < u1,2 < 1✓2,1 l2,1 < 2✓1,2 < u2,1 (3) FP(0,0) l1,2 < u1,2 < 1✓2,1 u2,1 < u2,1 < 2✓1,2 Hysteresis can be identiﬁed by tracking changes in Morse graphs over paths in parameter graph.
• 30. Signal control of the Toggle Switch 1 2S The rate of change of x1 is given by 1x1 + s · 1,2(x2) signal strength choice of logic We care about sign of 1✓2,1 + s · 1,2(x2) (7) FP(1,1) 1✓2,1 < sl1,2 < su1,2 2✓1,2 < l2,1 < u2,1 (8) FP(1,0) 1✓2,1 < sl1,2 < su1,2 l2,1 < 2✓1,2 < u2,1 (9) FP(1,0) 1✓2,1 < sl1,2 < su1,2 u2,1 < u2,1 < 2✓1,2 (4) FP(0,1) sl1,2 < 1✓2,1 < su1,2 2✓1,2 < l2,1 < u2,1 (5) FP(0,1) FP(1,0) sl1,2 < 1✓2,1 < su1,2 l2,1 < 2✓1,2 < u2,1 (6) FP(1,0) sl1,2 < 1✓2,1 < su1,2 l2,1 < u2,1 < 2✓1,2 (1) FP(0,1) sl1,2 < su1,2 < 1✓2,1 2✓1,2 < l2,1 < u2,1 (2) FP(0,1) sl1,2 < su1,2 < 1✓2,1 l2,1 < 2✓1,2 < u2,1 (3) FP(0,0) sl1,2 < su1,2 < 1✓2,1 u2,1 < u2,1 < 2✓1,2 DSGRN database Increasingsignals Use the product structure to count paths 1 4 7 2 5 8 3 6 9 Each graph gives rise to 6 possible monotone signal paths 1 ! 2 ! 3 1 ! 2 2 ! 3 21 3 Only one path 2 ! 5 ! 8 gives rise to hysteresis. 1 18 score:
• 31. Biological Model Biological Data/Phenotype ˙x = f(x, ) x 2 RN , 2 RM Physics/Math Model Traditional approach Finite Computational Model Part I Part II Part III Order theory Algebraic topology Choosing Models based on Robustness of Phenotype
• 32. What is the Phenotype? Signiﬁcance: Deregulation of the RB– E2F pathway is implicated in most, if not all, human cancers. Phenomena: Rb-E2F is a resettable bistable switch Bistability: Two equilibria: (A) E2F low = quiescence (B) E2F high = proliferation Resettable bistability: Bistable state: B When growth signals → 0 B → A A B S Hysteresis: A B S
• 33. Revisiting Yao et. al. DSGRN strategy Construct all subnetworks with 3 nodes satisfying the following properties: Every node has an out edge. There is at most one edge from one node to another node. Query product graphs over MD for resettable bistability and hysteresis. FP(MD,RP,EE) Quiescence:= FP(*,*,*,0) Proliferation:= FP(*,*,*,m)
• 34. Top choices of Yao, et. al. based on resettable bistability MD RP EE 21% 19% Hysteresis Resettable Bistability MD RP EE 17% 17% MD RP EE MD RP EE 8% 18% MD RP EE 8% 16% 6% 13% MD RP EE 4% 12% DSGRN Results