Solving x’=?

Solving .
Konstantin Mischaikow
Dept. of Mathematics, Rutgers
mischaik@math.rutgers.edu
Klagenfurt, June 2018
x0
=?<latexit sha1_base64="lhHaqzwYqs+x4iMBZ9v/sFdfYQE=">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</latexit><latexit sha1_base64="lhHaqzwYqs+x4iMBZ9v/sFdfYQE=">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</latexit><latexit sha1_base64="lhHaqzwYqs+x4iMBZ9v/sFdfYQE=">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</latexit><latexit sha1_base64="lhHaqzwYqs+x4iMBZ9v/sFdfYQE=">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</latexit>

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 sha1_base64="Rh9ijDcH0k3v5yQD+lTBVHGato4=">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</latexit><latexit sha1_base64="Rh9ijDcH0k3v5yQD+lTBVHGato4=">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</latexit><latexit sha1_base64="Rh9ijDcH0k3v5yQD+lTBVHGato4=">AAAZJXicpZlZbyO5EYC1m2vXuWaThxjICxHZyAygMSQjQYIgM1idtjyWj5Z8jC2v0U2xpR73BZK6prd/QX5InvOa/Ia8BQHylNf8jJDdtKbUJQ+QiQBbZH1VxaOLxSblxL4nZLX6r88+/853v/f9H3zx5dYPf/Tjn/z02Vc/uxTRlFN2QSM/4teOLZjvhexCetJn1zFnduD47Mp5aGp+NWNceFE4kMuY3QX2OPRcj9pSie6f7Q4zH4njT1m65T5fVPgL8oq49/z54sUrwhfPay8XL+6flat71exDcKFmCuWS+Zzdf/WL/wxHEZ0GLJTUt4W4rVVjeZfYXHrUVy0Np4LFNn2wxyzJepCSXSUaETfi6i+UJJOu6dmBCGw5SYtCrS4qqiCWgVMRMrD5ko+QGi2IlrmddsmFK9bg7VS6v79LvDCeShbSvHPu1CcyInoaycjjjEp/qQo25Z4aFaETm9tUqsleayZ2XiqdMbeDdGsNSO/hfd6oLvmew1W/E9Ud7i0qNufRXFTiSHj6QXnhuEJtn1bExI6ZKHiaMH/GpPLFWcjmNAoCOxwlQ9cOPH85Yq499WWaDIX7WN5as7/11YyNvfDV/l7VCyt8Vdv/rapKU61m0IG1u2TMooBJvtQed0m/PXhptY/rg3aL9OpN67S/NYQdUiNNb2t3ydBnrvyWJOVaOuTeeCK/TZFi5H5QHSZAd5gWlINZYMdqeBnN/uWTh32q4ae3+yunmc9K3plhJSnvmxb2nmjIGbmqmYkTLVRjAXFGO/fuTlFJTOP4yUGqOWqcHrfIcXswaFuF2XHsNFH/XWKjhh1DHESoIRR31pARImNDxohMDJkg4hniIfLOkHeIPBjygIhviI9IYEiASGhIiMjUkCkiM0NmiMwNmSOyMGSByNKQJSLvDXmf6pW5huoG1ZFRw5AGIk1Dmoi0DGkh0jakjUjHkA4iXUO6iPQM6SFiGWIh0jekj8jAkEGaL4HjevNN47RutT62GvLw0dnZcRIVQsWZrQNcT4ttNgBtINoEtIloC9AWom1A24h2AO0gegDogaa7a/gQ4ENk3AW0i+gRoEeIngB6gug5oOeIWoBaiDr5cze8j/gA0AGiF4BeIHoJ6CWiV4BeIXoN6DWibwF9i+gNoDdZ5KmobdaPuwdW/eyw29wcsXQVkWqnxuudNiBGi542IUYrn6qgfKQJWv20DY1RCqAdiFEeoAcQHyB8CPEhwl2IUSqhRxAfIfwG4jcIH0N8jHAPYpSs6AnEJwifQnyK8BnEZwifQ3yOsAUxypa0DzFKmXQA8QDhC4gvEL6E+BLhK4ivEL6G+BrhtxC/RfgG4ps84/ftkPQZ99zNK0esVo7YsFGKBqBo3YgmoGjZiBagaNmINqBo1YhDQFHYiy6gKOrFEaAo6MUbQFHMi2NAUciLHqAo4sUJoCjgxRmgKKDFOaAonoUFKApncQEoikdx+YEmBGdycQWsUTyKG0Bv0HuAsNgYuNc1E3SU/1oUdCn/MH2Ub5g/yjvpiicoVSrhAeAoVyphH3C0rpXQgh2w8HB6OhuDJ7wqHuB5a64rN9Mn7YrBG870YQw8FSNBTdT7a3pZFSudOJMR9JYLcAwV9azNepm5tcHhvZnC4oDcECh38BsOhbyJeQx7ddbK93wdRKRnx4IoHVLPDpWqXXWKVJpxdqYr11SHpDpGc+brk5469UXh+MMhVLdj9PVxcLO6JkXtUTQPtXbDGw91OeND7ttxsjOcURZKxs1ZdEdPiRdLIZc+K9fUiXQHOJrGj26m8Sc7CZl2ErJPdiAyB+L/cDDPHMw/fQiZg/B/d5B7yG4ktAdwdbC6JdglnYiTPzq+PXn9h1zfDseZtq0fb3YJQIY8q6yp1ypEf+2/zq089bDK+6YRYJtdThDT4LobHbNkHvGRIHNPTqKpJIEaTO4vG7Ouaif5a2yvPjgkW4XlQ32TLqI4/+ZBomTpMIx8L/CkKJ5U9O1C0UDJoMF6A5sM6McsuqHEFp4SAgtSNJlhEy18shFvQ6+8j3SqGYXYQAuhxbqJP+dg+zrNt6bj+mDQbbbRa5DMUq1Os3JDlrVYbLAubUiaj8abk2r3kXefSLoNP6IPuQ4xFXS55eRAbZIkuwRTEpWUq5Va+k0nb9wYos5D79Zm712o092o0+0/DuKJ/aoXccGMTl5O0y93idIlkUtsXx0TtFQLii8HvdajHT71i3hF4814RRUkeNqk/MicbXja77Ibt2CKnMVZPEm2kEnM2YbnOF1xMaUUcTu7fAvRZZneZTRZ7Tb4nfYdfKn9ZjhjanK37p+Va8VfAnDhcn+vVt2rnf+m/PVr8yvBF6Vfln5Vel6qlX5X+rp0WDorXZRo6U+lv5T+Wvrb9p+3/779j+1/5qqff2Zsfl5a+2z/+7+x65xJ</latexit><latexit sha1_base64="Rh9ijDcH0k3v5yQD+lTBVHGato4=">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</latexit>
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 sha1_base64="biMg1EkIecMtlHMDTfuHajxYUsc=">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</latexit><latexit sha1_base64="biMg1EkIecMtlHMDTfuHajxYUsc=">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</latexit><latexit sha1_base64="biMg1EkIecMtlHMDTfuHajxYUsc=">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</latexit><latexit sha1_base64="biMg1EkIecMtlHMDTfuHajxYUsc=">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</latexit>
N<latexit sha1_base64="rvDqvlscEPyNnNXLk7ttqBhLexE=">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</latexit><latexit sha1_base64="rvDqvlscEPyNnNXLk7ttqBhLexE=">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</latexit><latexit sha1_base64="rvDqvlscEPyNnNXLk7ttqBhLexE=">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</latexit><latexit sha1_base64="rvDqvlscEPyNnNXLk7ttqBhLexE=">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</latexit>
Attracting blocks are what we can hope to see from time series data.
N<latexit sha1_base64="DZ8SC50cVAmi4D0hOJf/qCa/psQ=">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</latexit><latexit sha1_base64="DZ8SC50cVAmi4D0hOJf/qCa/psQ=">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</latexit><latexit sha1_base64="DZ8SC50cVAmi4D0hOJf/qCa/psQ=">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</latexit><latexit sha1_base64="DZ8SC50cVAmi4D0hOJf/qCa/psQ=">AAAZGnicpZlLbyPHEYBpx05s2knW8cGCfWlEEpIDLZACggSBvTCfErWkHkPqsdLIwkyzSc5qXuhukuKOB8gPyTnX5DfkFvjqS/5Bfka6Z1rc4hS1QDYEJHbXV1X9mOrq6aYb+56Q1eq/33v/Zx98+PNffPRx+ZNPf/mrXz/77DcXIppxys5p5Ef8ynUE872QnUtP+uwq5swJXJ9duvdNzS/njAsvCodyGbPbwJmE3tijjlSiu2df2mHkhSMWyvKOnblLOBulxzvl8t2z7epeNfsQXKiZwnbJfE7vPvviP/YoorNAeaO+I8RNrRrL28Th0qM+S8v2TLDYoffOhCVZWynZVaIRGUdc/YWSZNI1PScQgSOnaVGo1UVFFcQycCtCBg5f8hFSowXRMrfTLrkYizV4M5PjP90mXhjPJAtp3rnxzCcyInruyMjjjEp/qQoO5Z4aFaFThztUqhleayZ2v1Y6E+4EaXkNSO/+dd6oLvmey1W/E9Ud7j1UHM6jhajEkfD00/HCSYU6Pq2IqRMzUfA0Zf6cSeWLs5AtaBQETjhK7LETeP5yxMbOzJdpYovxY7m8Zn/jqxmbeOG3+3tVL6zwVW3/D6oqTbWaQRfWbpMJiwIm+VJ73CWD9vBrq92rD9st0q83rZNB2YYdUiNNb2q3ie2zsfyBJNu11ObeZCp/SJFiNH6jaidA104LysE8cGI1vIxm//LJwz7V8NOb/ZXTzGcl74xdSbb3TQt7TzTkjsaqmakbPajGAuKOdu7GO0UlMYvjJwep5qhx0muRXns4bFuF2XGdNFH/x8RBDbuGuIhQQyjurCEjRCaGTBCZGjJFxDPEQ+SVIa8QuTfkHhHfEB+RwJAAkdCQEJGZITNE5obMEVkYskDkwZAHRJaGLBF5bcjrVK/MNVQ3qI6MGoY0EGka0kSkZUgLkbYhbUQ6hnQQ6RrSRaRvSB8RyxALkYEhA0SGhgzTfAn06s0XjZO61XrbasjDR2dn101UCBVntg5wPS222QC0gWgT0CaiLUBbiLYBbSPaAbSD6AGgB5ruruFDgA+RcRfQLqJHgB4hegzoMaJngJ4hagFqIermz93wAeJDQIeIngN6jugFoBeIXgJ6iegVoFeIvgT0JaLXgF5nkaeitlnvdQ+s+ulht7k5YukqItVOjdc7bUCMFj1tQoxWPlVB+UgTtPppGxqjFEA7EKM8QA8gPkD4EOJDhLsQo1RCjyA+QvgFxC8Q7kHcQ7gPMUpW9BjiY4RPID5B+BTiU4TPID5D2IIYZUs6gBilTDqEeIjwOcTnCF9AfIHwJcSXCF9BfIXwS4hfInwN8XWe8QdOSAaMe+PNK0esVo7YsFGKBqBo3YgmoGjZiBagaNmINqBo1YhDQFHYiy6gKOrFEaAo6MULQFHMix6gKORFH1AU8eIYUBTw4hRQFNDiDFAUz8ICFIWzOAcUxaO4eEMTgjO5uATWKB7FNaDX6D1AWIwC97pmgo7y34mCLuVvpo/yDfNHeSdd8QSlSiU8ABzlSiUcAI7WtRJasAMWHk5fZ2PwhFfFAzxvzXXlZvqkXTF4w7k+jIGnYiSoifpgTS+rYqVjdzqC3nIBjqGinrVZLzO3Nji8M1NYHNA4BMod/IZDIW9iHsNenbbyPV8HEek7sSBKh9SzQ6VqV50ilWacnem2a6pDUh2jOfP1SU+d+qJw8uYQqtsx+vo4uFldk6L2KFqEWrvhTWxdzrjNfSdOduw5ZaFk3JxFd/SUeLEUcumz7Zo6ke4AR7P40c0sfmcnIdNOQvbODkTmQPwfDhaZg8W7DyFzEP7vDnIP2Y2E9gCuDla3BLukE3Hyjes70+d/zvWdcJJpO/rxZpcAxOZZZU29ViH6a/95buWph7W9bxoBttnlBDENrrvRMUsWER8JsvDkNJpJEqjB5P6yMeuqdpK/xvbrw0NSLiwf6pt0EcX5Nw8SJUvtMPK9wJOieFLRtwtFAyWDBusNbDKgb7PohhJbeEoILEjRZI5NtPDJRrwNvfLe0qlmFGIDLYQW6yb+goPt6yTfmnr14bDbbKPXIJmlWp1m5YYsa7HYYF3akDQfjTcn1e4j7z6RdBt+RO9zHWIq6HLLzYHaJEl2CaYkKilXK7X0+07euDFEnYferc3eu1Cnu1GnO3gcxBP7VT/ighmdvJymH+8SpUuiMXF8dUzQUi0ovhz0W492+NQv4hWNN+MVVZDgaZPyLXO24Wm/ym7cghlyFmfxJNmDTGJ9P4+f0YqLGaWIO9nlW4guy/Quo8lqt8HvtK/gS+339pypydW/DdSKvwTgwsX+Xq26Vzvb3/7uufmV4KPSV6Xfln5fqpX+WPqudFg6LZ2XaOkvpb+V/l76x9Zft/659a+tH3PV998zNp+X1j5bP/0XrL2ZEQ==</latexit>

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
<latexit sha1_base64="phLODaoue346UCEO4Xqh76Rf3l0=">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</latexit><latexit sha1_base64="phLODaoue346UCEO4Xqh76Rf3l0=">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</latexit><latexit sha1_base64="phLODaoue346UCEO4Xqh76Rf3l0=">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</latexit><latexit sha1_base64="phLODaoue346UCEO4Xqh76Rf3l0=">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</latexit>
1. Experimentation
2. Comparison
S
Cln3
Whi5
SBF
Cln2
<latexit sha1_base64="Qh/qENagCLcx9/QZFxaxEDQMHIY=">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</latexit><latexit sha1_base64="Qh/qENagCLcx9/QZFxaxEDQMHIY=">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</latexit><latexit sha1_base64="Qh/qENagCLcx9/QZFxaxEDQMHIY=">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</latexit><latexit 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’=?

More Related Content

More from Förderverein Technische Fakultät

Recently uploaded

Solving x’=?