Dissecting Multiscale Systems Peter Szmolyan Technische Universit¨ at Wien, Austria
Introduction 1. Overview: time scales, slow-fast dynamics, model reduction 2. Olsen model: complicated dynamics 3. Cell cycle model: Mitotic Oscillator 4. Extensions to other regulatory or signaling networks 5. Conclusion and Outlook
1. Biological processes occur on widely di ff erent time scales and show slow-fast dynamics Rhythm Period Neural rhythms 0.01 - 1 s Cardiac rhythm 1 s Calzium-oszillations 1 s – min Biochemical oscillations 1 min – 20 min Cell cycle , signaling processes 10 min – 24 h Hormonal rhythms 10 min – 24 h Circadian rhythm 24 h Ovarian cycle 28 days periodic phenomena, A. Goldbeter (1996)
Chemical, metabolic, and gene regulatory networks are often modeled as systems of ODEs many variables many parameters very di ff erent orders of magnitude numerical simulation not too di ffi cult mathematical analysis challenging simple dynamics complicated dynamics many di ff erent time scales, slow-fast dynamics
Concept of slow manifolds is widely used ˙ dynamical system: X = F ( X ) , X 2 M all solutions rapidly approach lower-dimensional invariant slow manifold S ⇢ M , “fast modes are slaved to slow modes”, mathematical justification: center-, slow-, inertial-manifolds
Often several - hard to identify - slow manifolds exist Di ffi culties: several branches, di ff erent orders of magnitudes di ff erent dimensions, di ff erent types of stability, saddle type slow manifolds, bifurcations as parameters vary intersections? connections by fast dynamics?
Existing methods for dimension reduction need to be combined quasi steady state assumptions ad hoc reductions, lumping,... classical singular perturbation methods: expansions, Tikhonov theory geometric singular perturbation theory: invariant manifolds, Fenichel theory, blow-up method truncations based on tropical methods symbolic methods computational methods
Good scalings are crucial for model simplification and dimension reduction large terms dominate small terms finding a good scaling is nontrivial! what is a good scaling? nonlinear problem ) good scaling depends on position in phase space often there exist several good scalings
Good scalings are crucial for model simplification and dimension reduction large terms dominate small terms finding a good scaling is nontrivial! what is a good scaling? nonlinear problem ) good scaling depends on position in phase space often there exist several good scalings x 0 = � x + ε x + ε x 2 , x 2 R , ε ⌧ 1 ) x 0 = � x + O ( ε ) x = O (1) = x = O ( ε � 1 ) , x = X ) X 0 = � X + X 2 + O ( ε ) ε =
Most scalings have the form of power-laws x 0 = f ( x, µ ) , x 2 R n , µ 2 R m variables x , parameters µ reference value x = ξ , µ = m , often ξ = 0 , m = 0 + ε α i ˜ x i = x i , i = 1 , . . . , n ξ i µ j = m j + ε β j ˜ µ j , j = 1 , . . . , m key issue: find suitable ε ⌧ 1 , α 2 Q n , β 2 Q m tools: experience, intuition, Newton polygon, tropical methods, symbolic computations, numerical simulations,...
Regular perturbation problems are easy, singular perturbation problems are interesting A perturbation problem u = F ( u, ε ) , ˙ 0 < ε ⌧ 1 is regular , if it can be analyzed in a single scaling singular , if two or more di ff erent scalings are needed.
Singularly perturbed (slow-fast) ODEs in standard form require two scalings x = f ( x, y, ε ) ˙ (1) ε ˙ y = g ( x, y, ε ) x slow, y fast, ε ⌧ 1 , slow time scale t , transform to fast time scale τ := t/ ε x 0 = ε f ( x, y, ε ) (2) y 0 = g ( x, y, ε ) Syst. (1) and Syst. (2) equivalent for ε > 0
There are two distinct limiting systems for ε = 0 x = f ( x, y, 0) ˙ reduced problem 0 = g ( x, y, 0) chemistry: quasi steady state approximation x 0 = 0 layer problem y 0 = g ( x, y, 0) chemistry: fast approach to equilibrium critical manifold S := { g ( x, y, 0) = 0 } reduced problem is a dynamical system on S S is a “manifold” of equilibria for layer problem
Geometric Singular Perturbation Theory (GSPT) based on invariant manifold theory allows to go from ε = 0 to 0 < ε ⌧ 1 Theorem: critical manifold S normally hyperbolic, i.e. ∂ g ∂ y | S hyperbolic ) S perturbs smoothly to slow manifold S ε for ε small, ... N. Fenichel (1979)
Almost all slow-fast problems involve non-hyperbolic points folds of S self-intersections of S unbounded branches of S can be treated with blow-up method blow-up, well known in algebraic geometry rescaling + compactification simplest blow-up: polar coordinates
Slow-fast decompositions, GSPT + blow-up method have proven to be very useful enzyme kinetics mathematical neuroscience metabolic processes: glycolysis, oxidization of NADH cell cycle signaling pathways
2. Olsen model describes oxidization of Nicotinamide Adenine Dinucleotide (NADH) ! 2 NAD + + 2 H 2 O 2 NADH + O 2 + 2 H + � ˙ A = k 7 � k 9 A � k 3 ABY ˙ B = k 8 � k 1 BX � k 3 ABY X = k 1 BX � 2 k 2 X 2 + 3 k 3 ABY � k 4 X + k 6 ˙ = 2 k 2 X 2 � k 3 ABY � k 5 Y ˙ Y A oxygen, B NADH, X , Y intermediate products Reaction rates: k 1 = 0 . 16 , 0 . 35 , 0 . 41 k 2 = 250 , k 3 = 0 . 035 , k 4 = 20 , k 5 = 5 . 35 , k 6 = 10 � 5 , k 7 = 0 . 8 , k 8 = 0 . 825 , k 9 = 0 . 1
The Olsen model has complicated dynamics a) k 1 = 0 . 16 , b) k 1 = 0 . 35 , c) k 1 = 0 . 41 slow-fast dynamics: a) and b) mixed-mode oscillations or chaotic c) relaxation oscillations C. Kuehn and P. Sz., J. Nonlinear Science (2015)
Visualization in phase space shows more details slow dynamics in A, B close to X, Y ⇡ 0 , fast dynamics in A, B, X, Y away from X, Y ⇡ 0 ,
Scaling A, B, X, and Y gives a slow-fast system a = θ � α a � aby ˙ ˙ b = ν (1 � bx 2 � aby ) ( o ) ε 2 ˙ x = bx � x 2 + 3 aby � β x + δ ε 2 ˙ y = x 2 � y � aby ε 2 ⇡ 10 � 2 , ν ⇡ 10 � 1 , δ ⇡ 10 � 5 θ , α , β ⇡ 1 , ε ⌧ 1 = ) a, b slow x, y fast α ⇠ k 1 , δ ⇠ k 6 , bifurcation parameters
Finding the scaling is not easy p 2 k 2 k 8 k 1 k 5 A = a, B = p 2 k 2 k 8 b k 1 k 3 X = k 8 Y = k 8 x, y 2 k 2 k 5 k 1 k 5 T = p 2 k 2 k 8 t k 3 k 8 A. Milik (1996)
The Olsen model has a complicated critical manifold a = θ � α a � aby ˙ ˙ b = ν (1 � bx � aby ) ε 2 ˙ x = bx � x 2 + 3 aby � β x + δ ε 2 ˙ y = x 2 � y � aby For δ = 0 critical manifold S is defined by 0 = bx � x 2 + 3 aby � β x 0 = x 2 � y � aby solve for x, y Remark: δ 6 = 0 breaks invariance of x = 0 , y = 0
S has several branches and singularities ( x, y ) = (0 , 0) attracting b < β , saddle b > β and ( b � β ) 2 b � β x = 1 � 2 ab (1 + ab ) , y = (1 � 2 ab ) 2 (1 + ab ) attracting b > β , saddle b > β poles at 2 ab = 1 , transcritical singularities at b = β (non-hyperbolic)
GSPT explains dynamics for x, y = O (1) , ε ⌧ 1 a = θ � α a ˙ reduced flow in x = 0 , y = 0 ˙ b = ν return mechanism? needs x, y large!
Rescaling for x, y large explains return mechanism scaling x = X y = Y t = ε 2 τ ε , ε 2 , transforms System ( o ) to a 0 = ε 2 ( µ � α a ) � abY b 0 = ε 2 ν � ε bY � ν abY ( O ) ε X 0 = � X 2 + ε ( b � β ) X + 3 abY + ε 2 δ Y 0 = X 2 � Y � abY. a, b, Y slow, X fast
System (O) has a new three-dimensional attracting critical manifold C new critical manifold C := { X 2 = 3 abY } attracting, X > 0 non-hyperbolic at X = 0 , Y = 0 reduced problem on C is integrable a 0 = � abY b 0 = � ν abY Y 0 = (2 ab � 1) Y
Large amplitude returns occur on critical manifold C defined by 3 abY = X 2 GSPT ) rigorous results for ( X, Y ) 6 = (0 , 0) , ε ⌧ 1 Close to ( X, Y ) = (0 , 0) matching of System ( O ) to System ( o ) by blow-up method .
Blow-up method allows to match System ( O ) near ( X, Y ) = (0 , 0) and System ( o ) with x, y large Blow-up Transformation: include ε as a variable! r 2 ¯ ε ) 2 S 2 , ¯ X = ¯ r ¯ x, Y = ¯ y, ε = ¯ r ¯ ε , (¯ x, ¯ y, ¯ r 2 R weighted spherical blow-up of (0 , 0 , 0) in ( X, Y, ε ) space di ff eomorphism for ¯ r > 0 degenerate point (0 , 0 , 0) is blown-up to S 2 ⇥ { 0 } vector field can be pulled back to ¯ r = 0 and can be desingularized blow-up = rescaling + compacification
Two charts cover the relevant parts of the sphere S 2 r 2 ¯ blow-up: X = ¯ r ¯ x, Y = ¯ y, ε = ¯ r ¯ ε Chart K 1 defined by setting ¯ x = 1 , covers x > 0 X = r 1 , Y = r 2 1 y 1 , ε = r 1 ε 1 is equivalent to System ( O ) for r 1 > 0 Chart K 2 defined by setting ¯ ε = 1 , covers ε > 0 X = r 2 x 2 , Y = r 2 2 y 2 , ε = r 2 reproduces System ( o ) : X = ε x, Y = ε 2 y At ¯ r = 0 the system can be desingularized and rigorous matching near (¯ ε , ¯ r ) = (0 , 0) is possible.
Why does it work? ¯ C gets separated from the bad set (¯ x, ¯ y ) = (0 , 0) ; gain of normal hyperbolicity in parts of K 1 and K 2 we recover Systems ( O ) and ( O 2 ) which can be analysed by GSPT Within K 2 other blow-ups have to be used to study transcritical singularities ( δ = 0 ) or folds ( δ > 0 )
3. The mitotic oscillator is a simple model related to the dynamics of the cell-cycle cell-cycle: periodic sequence of cell divisions crucial players: Cyclin , Cyclin-dependent kinase (Cdk) complicated regulatory network Paul Nurse, Lee Hartwell, driven by an oscillator? Tim Hunt Nobel-Prize in medicine (2001)
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