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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
A Philosophical Interlude
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
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.
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)
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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Yeast START
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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

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Solving x’=?

  • 1. Solving . Konstantin Mischaikow Dept. of Mathematics, Rutgers mischaik@math.rutgers.edu Klagenfurt, June 2018 x0 =?<latexit 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  • 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
  • 7. 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?
  • 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}
  • 17. -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. 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  • 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
  • 36. 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 <latexit 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1. Experimentation 2. Comparison S Cln3 Whi5 SBF Cln2 <latexit 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Yeast START network
  • 37. 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