Mathematical Models of
Development in C. elegans
Dave Goulet
Department of Mathematics
Rose-Hulman Institute of Technology
Why C. Elegans ?
Gene
Human Protein Function
Worm
Ortholog
p53
Tumor Suppressor
cep-1
erbB-4
Epithelial Growth Factor Receptor
let-23
C. elegans
# cells
959
# base pairs ~ 1x108
# neurons 302
size (mm) ~ 1
H. sapiens
~ 1014
~ 3x109
~ 1011
~ 103
Successful Mating
Successful Mating
Unsuccessful Mating
Known Cell Lineage
Hermaphrodite Gonad Anatomy: Vulval Precursors
photos courtesy of wormatlas.org
Vulval Precursor Cell Induction
AC
Gonad
P3.p
P4.p
Hypodermis
P5p
P6.p
P7.p
6 will become 22
P8.p
Vulval Precursor Cell Induction
Vulval Precursor Cell Induction
Inductive Signal and Inhibitors
of the Map Kinase Cascade
lst-2
trafficking of EGFR
dpy-23
degradation of
EGFR
ARK-1
inhibit signaling
upstream of RAS
LIP-1
MAPK phosphatase
lst-1
directly on MAPK?
AC
LIN-3(EGF)
LET-23(EGFR)
SEM-5 (GRB-2)
LET-341
LET-60(RAS)
LIN-45 (RAF)
MEK-2
SUR-1(MAPK)
The Lateral Signal
cell
membranes
MAPK Activity
APX-1
transmembrane
LAG-2
transmembrane
DSL-1
secreted
LIN-12
(NOTCH)
Inhibitors of
MAPK
Cascade
Coupled Lateral and Inductive Signals
AC
lst-2
LIN-3
LET-23
dpy-23
SEM-5
LET-341
ARK-1
LET-60
APX-1
LAG-2
LIN-12
DSL-1
LIP-1
LIN-45
MEK-2
lst-1
lin-31, lin-39, and other transcription factors
SUR-1
Why Build a Mathematical Model?
0
What Promotes
Stable Fate
Outcomes?
1
2
3
4
5
6
7
8
9
Autocrine Signaling?
AC
lst-2
LIN-3
LET-23
DSL-1
(secreted)
LIN-12
dpy-23
SEM-5
LET-341
ARK-1
LET-60
LIP-1
LIN-45
MEK-2
lst-1
lin-31, lin-39, and other transcription factors
SUR-1
A Feedback Loop
DELTA
AC
NOTCH
I
N
H
I
B
I
T
O
R
TXN FACTORS
C
A
S
C
A
D
E
Vulval Precursor Cell Induction
Hermaphrodite Gonad Anatomy: AC and VU
Development of the Gonad
X. Karp, I. Greenwald / Developmental Bio
that
with
func
Mat
Gen
DARK = Terminally Differentiated
LIGHT = Gonad Primordium
Cell
Type
AC
Anchor Cell
VU
Ventral Uterine Precursor Cell
SS
Sheath/Spermatheca Cell
DTC
Distal Tip Cell
A
wise
Role
Direct Vulval Development eleg
Form Portion of Mature Uterus mut
1983
Promote Mitosis in Adjacent Cellssmg
pres
AC/VU Specification
Z1.ppp
Lateral
Signal
Z4.aaa
?
?
?
VU
AC
Karp & Greenwald
Active
LIN-12
hypothesized
Notch receptor
Z1.ppp
LIN-12
Notch receptor
Z4.aaa
“Factor-X”
post-transcriptional
down-regulator
HLH-2
helix-loop-helix
transcription promoter
LAG-2
transmembrane
DSL ligand
Why Build a Mathematical Model?
Observations:
Z1.ppp/Z4.aaa become AC by default in the absence of lateral signal.
LIN-12 activation leads to the VU fate.
Decision seems arbitrary, except that the first born usually becomes VU.
Birth time separations 2min-2hr.
Problems:
Why does first born become VU?
Does this model provide for the rapid amplification of small differences?
Coupled Chains
LAG-2
HLH-2
Factor-X
LIN-12
?
LIN-12
?
Factor-X
HLH-2
LAG-2
Coupled Chains
Is there a common way to model these two situations?
MapK Pathway
AC
lst-2
LIN-3
LET-23
dpy-23
SEM-5
LET-341
ARK-1
LET-60
APX-1
LAG-2
LIN-12
DSL-1
LIP-1
LIN-45
MEK-2
lst-1
lin-31, lin-39, and other transcription factors
SUR-1
Abstraction
External
Signal
Bound Ligand
Internalized Ligand
Intracellular Cascade
Nuclear Signal
DNA Transcription
RNA Translation
Protein Trafficking
Secreted Protein
Example Kinetics
The Author
A
B
April 12, 2005
dA
=
dt
Law of Mass Action
A+E
dA
Michaelis-Menten Kinetics
with Fast Equilibrium
dt
=
k+ A + k B
C
k+ A + k B
dB
=
dt
k2 E0 A
k1 +k2
k1+
+A
B+E
Modeling Vulval Development
AC
lst-2
LIN-3
LET-23
dpy-23
SEM-5
LET-341
ARK-1
LET-60
APX-1
LAG-2
LIN-12
DSL-1
LIP-1
LIN-45
MEK-2
lst-1
lin-31, lin-39, and other transcription factors
SUR-1
Real ≠ Ideal
Ideal:
“...(parameter estimates) are essential in any practical
application of a model to a specific biological problem”
-J.D. Murray*
Real:
Parameters and mechanisms are often unknown
* Mathematical Biology II: Spatial Models and Biomedical Applications, pg 417, Springer 2003
Modeling Isolated VPC
for 0 ≤ x < xi
c u(0, t) = s(t)
u(x, 0) = 0
−
+
c u(x+
i , t) = c u(xi ) − qi u(xi , t) v(t)
at x = xi
dv
= c u(xe , t) − qe u(x+
i , t) v(t)
dt
v(0) = v0
for xi < x ≤ xe
∂u
∂u
+c
=0
∂t
∂x
P
∂u
∂u
+c
=0
∂t
∂x
A Reduced Model
V (τ ) H(τ − φ)
dV (τ )
H(τ − 1)
=
− βe
dτ
1 + βi V (τ − 1 + φ))
1 + βi V (τ )
V (τ ) = γ
H : Heaviside function
for τ ∈ [−1, 0]
1
U (τ ) =
1 + βi V (τ + φ)
V(τ)
~
Inhibitor level as a function of time
U(τ)
~
MAPK activity as a function of time
τ
~
time
β
γ
ϕ
~
signal strength*inhibitor strength ÷ (signal transduction speed)^2
~
initial level of inhibitor ÷ inductive signal strength
~
relative point of action of the inhibitor
Different Inhibitors
lst-2
dpy-23
ark-1
Increasing
Signal
Strength
MapK Activity vs. Time
lip-1
lst-1
Inhibitor Strength
Increasing
Inhibitor Strength
Increasing
Signal
Strength
MapK Activity vs. Time
Transduction Speed
Increasing
Speed
Increasing
Signal
Strength
Notch Activity vs. Time
Modeling AC/VU Specification
Z1.ppp
Lateral
Signal
Z4.aaa
?
?
?
VU
AC
Coupled Chains
LAG-2
HLH-2
Factor-X
LIN-12
?
LIN-12
?
Factor-X
HLH-2
LAG-2
Coupled Chains
✓
◆
dt
N
±
kN
= k±
! k ± (x)
M
The Model ✓
◆
dX2 (t)
@u
@
@
@u
+
(c u) =
D
+ O( x)
dt
@t
@x
@x
@x
dX1 (t)
dt
dX2 (t)
dt
=
⌘ H(t 1)
⌘ + X2 (t 1)
X1 (t)
1 + X1 (t)
=
H(t ⌘)
1 + X1 (t ⌘)
X2 (t)
⌘ + X2 (t)
X1 (t) =
1
for t 2 [ 1, 0]
=
=
⌘+
X2 (t
(6)
1)
1+
H(t ⌘)
(7)
1 + X1 (t ⌘)
⌘+
1
L1 (t) =(8)
1 + X1 (⌧ )
(9)
⌘
L2 (t) =
(10)⌘ + X2 (⌧ )
1
X
(t)
=
for
t
2
[
⌘,
0]
2
2
X(t)
~
Factor-X
1
L(t)
L
~
1 (t)
lag-2
transcription
=
(11)
1 + X1 (⌧ )
⌘
β
L
~
2 (t)
strength
of
inhibition
÷
signal transduction speed(12)
1
=
⌘ + ofX
2 (⌧ )
γ
~
initial level
Factor-X
÷ HLH-2 production rate 1
η
~
signal transduction speed ratio
1
σ
~
HLH-2 production rate ratio
Solutions
LAG−2 Expression Rate vs. Time
δ=0.1
1
δ=1.1
δ=2.1
δ=3.1
0.8
δ=4.1
1/(1+Wi(t))
δ=5.1
δ=6.1
0.6
0.4
0.2
0
0
10
20
30
40
50
60
70
80
90
100
t
small birth time separation leads to
small differences in gene expression
Solutions Proposed by Karp & Greenwald
Prop. 1:
There is another promoter of lag-2 early after Z1.ppp and Z4.aaa are born.
Prop. 2:
Early on HLH is present below detection levels and the Z1 are very sensitive to it.
☹
More LAG-2
More likely to be AC
Prop. 3:
LIN-12 accumulates from birth giving the first born more receptor when the interactions begin.
Prop. 4:
Another source of LAG-2 signals the first born to create a slight bias.
☹
Plots show that initial peaks are overcome in the long term.
☹
Even when cells were born <10min apart, the first born became VU.
☹
Small differences in Factor-X small differences in LAG-2.
Prop. 5:
Other unknown pathways influence LIN-12 activity.
☺" What are these pathways?
Solution Proposed by Me
Prop. 6:
HLH is promoted at different rates in the two cells due to inherent differences in the two cells.
☹
This violates the equipotential hypothesis.
☺" It works!
Lessons From Drosophila
Active
Notch
Pre-pattern
Factors
Active
Notch
E(spl)
SMC
AS-C Enhancer
Delta
Proposition 7
Active
LIN-12
FactorX
?
Active
LIN-12
HLH-2
LAG-2
?
Solutions
LAG−2 Expression Rate vs. Time
δ=0.1
1
δ=1.1
δ=2.1
δ=3.1
0.8
δ=4.1
1/(1+Wi(t))
δ=5.1
δ=6.1
0.6
0.4
0.2
0
0
5
10
15
20
25
30
35
40
45
50
t
small birth time separation leads to
BIG differences in gene expression
End
Destructive Inhibition
βi = βe
Equilibrium is stable: 1/β
Oscillatory modes decay more slowly for: (1 − φ)β/4 > 0.28 . . .
Dimensional:
x
Α
0.28
2 s∗ /c
!
"
xi qi s∗ xe
1−
> 1.113
2
xe
c
-1.28
-2
-3
-4
-5
1
2
3
4
HLH-2 Production: Strong Amplification
First Born
Becomes
AC ?
lag-2 Transcription vs. Time
Initial Factor X Level: Weak Amplification
First Born
Becomes
VU !
lag-2 Transcription vs. Time
Competing Effects ?
Lower HLH-2 production
VU
Higher Initial “Factor-X”
VU
Other Explanations ?
“...hlh-2 appears to be transcribed in both Z1.ppp and Z4.aaa...” - K&G, 2003
No mention of differences, but slight
differences do get multiplied!
Contrasting results from 2003 &
2004 suggests that the temporal
expression of HLH-2 is not well
understood.
Small differences likely exist.
Other Negative Feedback ?
Active
LIN-12
Notch receptor
?
transcription
activator
LIN-12
Notch receptor
“Factor-X”
post-transcriptional
down-regulator
HLH-2
transcription
promoter
LAG-2
transmembrane
ligand
Correct Fate Specification
DDE Asymptotics
Result: For ε≪1, under certain restrictions on the vector valued functions F, G and H, the following
non-linear system of DDE initial value problems
!
"
!
dV
! (t), V
! (t − 1), t
= " F! V
dt
! (t) = G(t,
! ") f or t ∈ [−1, 0)
V
! (0) = H(")
!
V
has solutions uniformly approximated to O(ε) by
! (t) ∼ X
! (η) + # Y (η)
V
η
!
!−
A
"
0
η
#
$ %
! X(s),
!
!
Y −1 (s) B
∇X(s)
ds
= # (t − 1)
where X solves a non-linear ODE system,Y is the fundamental solution matrix of a linear ODE
system, B is a functional depending on the form of F, and A is a constant depending on F, G, and H.
1
Linear Chains
The Author
April 12, 2005
A
A
N-1
dAN
+
= kN
dt
1 AN
1
A
N
+
(kN
+ kN ) AN + kN +1 AN +1
dB
=
dt
⇢
N
1 2Analysis:
M
A Lesson from Numerical
2
, ,...,
M
M M
N+1
M
k2 E0 A
k1 +k2
k1+
+A
! x 2 [0, 1]
Difference Scheme + 1Taylor
! Series
x
↓ M
PDE + numerical dispersion/diffusion term
Z
(N + 1 )/M
Z
(x+ 1 ) x
Continuum Model of
dAN
+
=
k
N 1 AN
Linear
Chains
dt
1
+
(kN
+ kN ) AN + kN +1 AN +1
dB
k2 E0 A
dAN
+
+
=
=dA
kNN 1 AN+ 1 (kN + kN+) AN + kN +1 AN +1
(1)2
k
+k
dt
1
dt
+ A(1
= kN 1 AN 1 (kN + kN ) AN + kN +1 AN +1
+
k
1
dt
dB
k2 E0 A
= dB
(2)
k
E
A
2
0
k
+k
2
dt
1
=
(2
+
⇢dtk1 +k1A+k2
N
1 2 k+ +MA
# of AN = M → ∞
2
, , 1. . . ,
! x 2 [0, 1]
M
M M
M
⇢
1
N
1 2⇢
M
!
x (3)
2 N , , 1. . . , 2
!
x 2 [0, 1]
M
M
M
M 2M
, M, . . . ,
! x 2 [0, 1]
(3
M
M M 1
M
!1 x
(4)
M
x
(4
1
Z (N!
Z (x+ 1 ) x
+
)M
M
2
2
Define a Continuous Variable
AN (t) =
u(s, t) ds !
u(s, t)
(N 12 )M !
(x 12 ) x
1
1 (N + 1 )/M
1
Z (N +!
Z
x+
)M
(x+ 2 ) x
✓ 2 ∆x
◆
2
2
1
1
Z
Z
(Nu(s,
+ 2 )/M
xds u(s, t)ds
± (x+
±2 )t)N
±
AN (t)A=
t) dsu(s,
!t)ds k→
u(s,
(5)
N (t) =
=
k
!
k
(x)
1
1
N
(N =2 )M
AN (t)
u(s, t) ds
(5
(N − 12 )/M u(s, t) ds (x !2 ) x
x− 12M
∆x
1✓
1
(N 2 )/M ◆
(x 2 ) x
N
✓! ◆k ± (x)
±
kN
= k± ±
(6)
N
±
±
Find an Approximate PDEk =
Mk
✓
◆
!
k
(x)
(6
N
!
"
@u
@
@
@u
M
+∂u (cu) =
D
∂
∂
∂u
2
@tD @x + O(∆x)
@x
@x
+
(c u) = ∆x
✓
◆
∂t @∂x
∂x
∂x
@u
@
@u
✓
◆
dt
Boundary Conditions ⇢
N
2
M
=
Advection-Diffusion Equation
(N + 2 )/M
(N
1
)/M
2
±
kN
(2)
+A
1 2
M
, ,...,
M M
M
e.g.
Specify Flux into First State
Specify Zero Flux out of LastZ State1
AN (t) =
k1 +k2
k1+
1
M
! x 2 [0, 1]
(3)
!
(4)
u(s, t) ds !
N
M
◆
Z
x
x+ 12
1
2
x
u(s, t) ds
dX (t)
dt
=
⇥ ⇤ H(t 1)
⇥ + X2 (t 1)
u
+dX2 (t)
(c u) = 0 H(t ⇥)
t
x
=
dt
1 + X1 (t ⇥)
dV
= (c u)x = xend
dt
1
L (t) =
(5)
x
=k
! k ± (x)
✓
◆
⌥u
⌥
⌥
⌥u
+
(c u) =
D
+ O( x)
⌥t
⌥x
⌥x
⌥x
1
Advection Equation
±
✓
x
(6)
☺
(7)
X1 (t)
1 + X1 (t)
(8)
⇤ X (t)
☹
2
⇥+
X2 (t)
(9)
(14)
(10)
(15)
(11)
Boundary Conditions
c u = ξ(t)
x=0
∂u
∂u
+c
=0
∂t
∂x
1 !
x"
u= ξ t−
c
c
!
1
cu = ξ t −
c
x=1
!
1
dV
=ξ t−
dt
c
"
"
Continuous vs. Discrete: 5=∞ ?
M=100
k+/k- = 20
+
k /k =
1.05
M=5
Rate Limiting Step
M=20