Solving x’=?

Solving .
Konstantin Mischaikow
Dept. of Mathematics, Rutgers
mischaik@math.rutgers.edu
Klagenfurt, June 2018
x0
=?<latexit 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Context/Motivation
(There are different concepts of
solving a differential equation)

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

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.

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

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

The equivalence relation:
Two maps f : X ! X and g: Y ! Y are topologically conjugate if
there exists a homeomorphism h: X ! Y such that h f = g h.
0 2 ⇤ is a bifurcation point if for any neighborhood U of 0 there
exists 1 2 U such that f 0 is not conjugate to f 1
The places of change:
f(x, r) = fr(x) = rx(1 x)<latexit 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Worth Noting:
Bifurcations are
occurring on all scales.
How does this help
our understanding?

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

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

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.

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

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
= . . .

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

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:

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}

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
A Definition from Continuous Dynamics
-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
Let ': R ⇥ X ! X be a ﬂow. A set N ⇢ X is an attracting block if
'(t, cl(N)) ⇢ int(N) for all t > 0.<latexit 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Attracting blocks are what we can hope to see from time series data.
N<latexit 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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))

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

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

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

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

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

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)

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)

✓2,1
✓1,2
x1
x2
Need to Construct State Transition Graph Fz : X !! X
Example (The Toggle Switch) 12
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.

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}
Constructing state transition
graph Fz : X !! X
Check signs of i✓j,i + i,j(xj)

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

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.

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:

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

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

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)

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

Extending Minimal Models
S
Myc
CycD
Rb
E2F
CycE
2a
2b8 7

Myc
CycD
Rb
E2F
CycE
2a
2b8 7
How do we check our
solution?
12.8%
23.81%
Hysteresis
(full path)
Resettable
Bistability
(full path)
S
Myc
CycD
Rb
E2F
CycE
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1. Experimentation
2. Comparison
S
Cln3
Whi5
SBF
Cln2
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sha1_base64="Qh/qENagCLcx9/QZFxaxEDQMHIY=">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</latexit>
Yeast START
network

Thank-you for your Attention
Homology + Database Software
chomp.rutgers.edu
Rutgers
S. Harker
MSU
T. Gedeon
B. Cummings
FAU
W. Kalies
VU Amsterdam
R. Vandervorst

Solving x’=?

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Solving x’=?