Coarse Graining via Polynomial Chaos Expansion:
Circadian Rhythms in the Suprachiasmatic Nucleus
Tom Bertalan
15 May, 2014
May 17, 2014
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Outline
Outline
Heterogeneous network systems
Kuramoto coupled oscillators
Hodgkin-Huxley neurons
Coarse dynamics
Coarse variables
Coarse projective integration
Polynomial chaos expansion
(PCE)
Multidimensional PCE basis
functions
Circadian rhythms
Biology
Gene expression model
Intercellular signaling
Signals
Topology
Results
bibliography
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Heterogeneous network systems
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Heterogeneous network systems
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Heterogeneous network systems
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Heterogeneous network systems
Kuramoto coupled oscillators
Kuramoto coupled oscillators
N
X
˙θi = ωi + K
Ai,j sin(θj − θi )
N
j=1
ω ∼ N (µ, σ)
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Heterogeneous network systems
Kuramoto coupled oscillators
(video: slowmoving.mp4)
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Heterogeneous network systems
i
C dV
dt
dmi
dt
dhi
dt
dni
dt
dsi
dt
Hodgkin-Huxley neurons
= I − gNa mi3 hi (Vi − VNa ) − gK ni4 (Vi − VK ) − gl (Vi − Vl ) + Isyn,i
i
= m∞τm(V(Vi )−m
i)
=
=
=
h∞ (Vi )−hi
τh (Vi )
n∞ (Vi )−ni
τn (Vi )
1−si
− sτi
1+e −Vi /5
f∞ (Vi ) = ax (Vi )/(ax (Vi ) + bx (Vi ))
τx (Vi ) = 1/(ax (Vi ) + bx (Vi ))
P
Isyn,i = − Ng N
j6=i Ai,j sj (t)[Vi (t) − Vsyn ].
ax and bx are various exponential and sigmoidal functions of V .1
1
Sung Joon Moon et al. “Coarse-grained clustering dynamics of synaptically coupled heterogeneous neurons”. Submitted
(2014).
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Coarse dynamics
Coarse projective integration
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Coarse dynamics
Polynomial chaos expansion (PCE)
For the response variable y and explantory variable x ∼ ρ(x), we seek the
expansion
M
X
yi ≈ yˆi =
αj φj (xi ),
j=1
where the orthonorality of the basis is shown by
δj,k =
hφj (x), φk (x)iρ
,
hφj (x), φj (x)iρ
and the inner product is defined as
Z
hφj (x), φk (x)iρ =
φj (x)φk (x)ρ(x)dx.
x∈Ω
The density ρ is the PDF according to which the {xi } sample is drawn
from the space Ω. For continuous functions, the coeffecients can be found
by an inner product
αj = hy (x), φj (x)iρ .
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Distribution
Gamma
Beta
Poisson
Gaussian
Uniform
Negative binomial
Binomial
Hypergeometric
Polynomials2
Laguerre
Jacobi
Charlier
Hermite→
Legendre
Meixner
Krawtchouk
Hahn
Polynomial chaos expansion (PCE)
4
2
φi (x)
Coarse dynamics
0
1
2
4
2.0
1.5
1.0
x
x2 −1
x3 −3x
x4 −6x2 +3
0.5 0.0
x
0.5
1.0
1.5
2.0
The first few probabalists’ Hermite
polynomials, orthogonal with respect to
2
ρ(x) = e −x /2 .
2
Dongbin Xiu and George Em Karniadakis. “The Wiener–Askey polynomial chaos for stochastic differential equations”.
SIAM Journal on Scientific Computing 02912.2 (2002).
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Coarse dynamics
Polynomial chaos expansion (PCE)
For two uncorrelated heterogeneities with weightings ρa (xa ) and ρb (xb ),
We can create a 2D basis as the Cartesian product of the 1D bases
Φl=j,k = φj (xa )φk (xb ).
Since the heterogeneities are independent,
R(xa , xb ) = ρa (xa )ρb (xb ),
so the orthogonality integral can be separated, and we can use the former
scheme for developing bases.
RR
δlm ∼ hΦl , Φm iR = RR Φl (xa , xb )Φm (xa , xb )dxa dxb
= R φla (xa )φlb (xb ) · φma (xRa )φmb (xb )dxa dxb =
φla (xa )φma (xa )dxa
φlb (xb )φmb (xb )dxb
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Coarse dynamics
Polynomial chaos expansion (PCE)
PCE on Kuramoto
N
X
˙θi = ωi + K
Ai,j sin(θj − θi )
N
j=1
ω ∼ N (µ, σ) Ai,j = 1
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Coarse dynamics
Polynomial chaos expansion (PCE)
PCE on Kuramoto with heterogeneous network
N
KX
Ai,j sin(θj − θi )
θ˙i = ωi +
N
j=1
Ai,j created by a random process, such that degree di =
in distribution.
P
j
Ai,j converges
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Coarse dynamics
Polynomial chaos expansion (PCE)
PCE on Hodgkin-Huxley
PCE for state h in a simulation of 128 spiking Hodgkin-Huxley neurons,
heterogeneous in membrane capacitance C and potassium reversal
potential VK .
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Coarse dynamics
Polynomial chaos expansion (PCE)
(video: tiled.mp4)
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Circadian rhythms
I
I
Biology
Bialateral suprachiasmatic
nucleus (SCN) of the mammal
hypothalamus (Karatsoreos
et al. 2004).
Core gene expression model3
has 16 ODEs.
3
Jean-christophe Leloup and Albert Goldbeter. “Toward a detailed computational model for the mammalian circadian clock”.
Proceedings of the National Academy of Sciences of the United States of America 100.12 (2003).
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Circadian rhythms
Gene expression model
mRNA ODEs (PER, CRY, and BMAL1)
Key: (variables), (parameters).
d(MP)
dt
= −(kdmp)(MP) −
(vmP)(MP)
(KmP)+(MP)
+
d(MC )
dt
= −(kdmc)(MC ) −
(vmC )(MC )
(KmC )+(MC )
+
d(MB)
dt
=
(KIB)(m) (vsB)
(KIB)(m) +(BN)(m)
(np)
((L)+1)
(vsP)(BN)
(KAP)
(np)
(np)
+(BN)
(nc)
(vsC )(BN)
(nc)
(KAC )
− (kdmb)(MB) −
+(BN)
(nc)
(vmB)(MB)
(KmB)+(MB)
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Circadian rhythms
Gene expression model
Per and Cry protein ODEs
includes phosphorylated, nuclear, and complexed variants
(V 1P)(PC )
(V 2P)(PCP)
(V 1C )(CC )
(V 2C )(CCP)
d(PC )
dt
=
− (Kp +(PC ) + (Kdp +(PCP) − (k3)(CC )(PC ) + (k4)(PCC ) − (kdnp) (PC ) + (ksP)(MP)
p)
p)
d(CC )
dt
=
− (Kp +(CC ) + (Kdp +(CCP) − (k3)(CC )(PC ) + (k4)(PCC ) − (kdnc)(CC ) + (ksC )(MC )
c)
c)
d(PCP)
dt
=
(V 1P)(PC )
(Kpp) +(PC )
− (Kdp +(PCP) − (kdnpp) (PCP) − (Kd +(PCP)
p)
p)
d(CCP)
dt
=
(V 1C )(CC )
(Kpc) +(CC )
− (Kdp +(CCP) − (kdncp) (CCP) − (Kd +(CCP)
c)
c)
d(PCC )
dt
=
− (Kp
(V 2P)(PCP)
(vdPC )(PCP)
(V 2C )(CCP)
(vdCC )(CCP)
(V 1PC )(PCC )
pcc) +(PCC )
(V 2PC )(PCCP)
+ (Kdp
− (k1)(PCC ) + (k2)(PCN) + (k3)(CC )(PC )
pcc) +(PCCP)
−(k4)(PCC ) − (kdnpcc) (PCC )
d(PCN)
dt
=
(V 3PC )(PCN)
pcn) +(PCN)
− (Kp
(V 4PC )(PCNP)
+ (Kdp
+ (k1)(PCC ) − (k2)(PCN) − (k7)(BN)(PCN)
pcn) +(PCNP)
+(k8)(IN) − (kdnpcn) (PCN)
d(PCCP)
dt
=
(V 1PC )(PCC )
(Kppcc) +(PCC )
− (Kdp
− (kdnpccp) (PCCP) − (Kd
pcc) +(PCCP)
pcc) +(PCCP)
(V 2PC )(PCCP)
(vdPCC )(PCCP)
d(PCNP)
dt
=
(V 3PC )(PCN)
(Kppcn) +(PCN)
− (Kdp
− (kdnpcnp) (PCNP) − (Kd
pcn) +(PCNP)
pcn) +(PCNP)
(V 4PC )(PCNP)
(vdPCN)(PCNP)
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Circadian rhythms
Gene expression model
Bmal1 protein and inactive complex ODEs
includes phosphorylated, nuclear, and complexed variants
(V 1B)(BC )
bc) +(BC )
d(BC )
dt
=
d(BCP)
dt
=
(V 1B)(BC )
(Kpbc) +(BC )
d(BN)
dt
=
− (Kp
− (Kp
(V 2B)(BCP)
+ (Kdp +(BCP) − (k5)(BC ) + (k6)(BN) − (kdnBc) (BC )
bc)
+(ksB)(MB)
(V 2B)(BCP)
(vdBC )(BCP)
− (Kdp +(BCP) − (kdnbcp) (BCP) − (Kd +(BCP)
bc)
bc)
(V 3B)(BN)
bn) +(BN)
(V 4B)(BNP)
+ (Kdp +(BNP) + (k5)(BC ) − (k6)(BN) − (k7)(BN)(PCN)
bn)
+(k8)(IN) − (kdnbn) (BN)
(V 4B)(BNP)
(vdBN)(BNP)
d(BNP)
dt
=
(V 3B)(BN)
(Kpbn) +(BN)
d(IN)
dt
=
(k7)(BN)(PCN) − (k8)(IN) − (kdnin) (IN) − (Kd +(IN)
in)
− (Kdp +(BNP) − (kdnbnp) (BNP) − (Kd +(BNP)
bn)
bn)
(vdIN)(IN)
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Circadian rhythms
Intercellular signaling
To et al. 20074 add one more differential equation for phosphorylated
cAMP response element binding (CREB) protein, (CB ∗ ), allowing for
intercellular vasoactive intestinal peptide (VIP) signalling. For N cells, the
number of equations rises from 16 to N · 17.
Key: (variables), (parameters), (algebraic equations).
d(MP)
dt
d(CB ∗ )
dt
(np)
(L+1)
(vmP)(MP)
+ (vsP)(BN)
= −(kdmp)(MP) − (KmP)+(MP)
(np)
(n
(KAP)
+(BN)i p)
h
∗
∗
(vP)
(vK )
1−(CB )
(CB )
=
(CBT )
(vP) (K 1)+(1−(CB ∗ )) − (K 2)+(CB ∗ )
∗
)(CB )
(vsp) = (vsp0) + (CT ) (KC(CBT
)+(CBT )(CB ∗ )
(CaC )
(vK ) = (VMK ) (Ka)+(CaC
)
(CaC ) = ((v 0) + (v 1)(β) + (v 2)(L))/(k)
(β) = (γ)/((KD) + (γ)
(γi ) =
(ρi ) =
PN
j=1 (αi,j )(ρj )
PN PN
i=1
j=1 (αi,j )/N
(MPi )
(a) (MPi )+(b)
4
Tsz-Leung To et al. “A molecular model for intercellular synchronization in the mammalian circadian clock.”
Biophysical Journal 92.11 (June 2007), pp. 3792–3803.
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Circadian rhythms
Intercellular signaling
Cells are embedded in 2D physical
space and connected in an
undirected graph with pairwise
weights (αi,j ) inversely proportional
to their distances.
In future work, distinct signaling
topologies will be used for the
anatomically separate core and shell
regions of the SCN.
Figure from To et al. 20075 .
5
Tsz-Leung To et al. “A molecular model for intercellular synchronization in the mammalian circadian clock.”
Biophysical Journal 92.11 (June 2007), pp. 3792–3803.
May 17, 2014
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Circadian rhythms
netIntegrate1400125128
7
L
0.10
5
t =128.0 [hr]
2.5
0.08
0.06
3
0.04
2
MP
MC 0.02
MB
1
20
40
60
80
time [hr]
100
120
0.00
140
(a) The three mRNA trajectories.
2.0
MP [nM]
4
0
0
0.12
light level L
mRNA concentration [nM]
6
Results
O(2) Hermite fit
data
1.5
1.0
0.5
0.0
1.50
1.55
1.60
vsP0 [nM h−1 ]
1.65
1.70
1.75
(b) A fit with O(3) Legendre
polynomials.
Square-wave light forcing with uniformly-distributed vsP0 (basal rate of PER
transcription).
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Circadian rhythms
Results
(video: images/netIntegrate1400125128.mp4)
May 17, 2014
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Conclusion
Heterogeneous network systems
Kuramoto coupled oscillators
Hodgkin-Huxley neurons
Coarse dynamics
Coarse variables
Coarse projective integration
Polynomial chaos expansion
(PCE)
Multidimensional PCE basis
functions
Circadian rhythms
Biology
Gene expression model
Intercellular signaling
Signals
Topology
Results
bibliography
May 17, 2014
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bibliography
Ilia N Karatsoreos et al. “Phenotype matters: identification of
light-responsive cells in the mouse suprachiasmatic nucleus.”
Journal of Neuroscience 24.1 (Jan. 2004), pp. 68–75. arXiv: 48.
Jean-christophe Leloup and Albert Goldbeter. “Toward a detailed
computational model for the mammalian circadian clock”.
Proceedings of the National Academy of Sciences of the United States o
100.12 (2003).
Sung Joon Moon et al. “Coarse-grained clustering dynamics of
synaptically coupled heterogeneous neurons”. Submitted (2014).
Tsz-Leung To et al. “A molecular model for intercellular
synchronization in the mammalian circadian clock.”
Biophysical Journal 92.11 (June 2007), pp. 3792–3803.
Dongbin Xiu and George Em Karniadakis. “The Wiener–Askey
polynomial chaos for stochastic differential equations”.
SIAM Journal on Scientific Computing 02912.2 (2002).
May 17, 2014
26 / 26
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