MMsemester project

CELLULAR OSCILLATIONS DUE TO
CONTRACTILITY AND TURNOVER
A semester project report Submitted
in Partial Fulﬁlment of the Requirements
for the course
Mathematical Methods
by
PREETI SAHU
to the
Physics Dept.
Syracuse University
30th
December 2015

Abstract
Collective and single-cell oscillations are observed in numerous bio-
logical systems[1]. In many of them, these processes are driven by the
actomyosin cytoskeleton. In this project we introduce turnover into an
otherwise simplistic model of a cell (comprising just of elastic springs,
dashpots and/or a constant conractile force) to observe changes in the
stability of the ﬁxed point.
2

Contents
1 Introduction 4
1.1 Types of Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.1.1 Newtonian liquid drop model . . . . . . . . . . . . . . . . 5
1.1.2 Perfectly Elastic . . . . . . . . . . . . . . . . . . . . . . . 5
1.1.3 Maxwell liquid drop model . . . . . . . . . . . . . . . . . 6
1.1.4 Linear viscoelastic solid model . . . . . . . . . . . . . . . 6
2 A realistic cell 7
2.0.5 Case-I [t < 1] . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.0.6 Case-II [t > 1] . . . . . . . . . . . . . . . . . . . . . . . . 8
3 Introducing Turnover 9
A Appendix A 13
3

1 Introduction
There are many scales that one can chose to study the working of a living cell.
Figure 1: Image Courtesy: Ref-3, Mechanical models for living cells- A review
The highlighted one, is the route we want to follow. After delving into
some of these models, chronologically with increasing complexity, one can easily
discover the most realistic one amongst them. In subsequent pages, we shall go
through elementary calculations to get a feel of their behaviour.
4

1.1 Types of Models
1.1.1 Newtonian liquid drop model
For a completely viscous ﬂuid1
, a constant external force for some time, just
stretches the cell irreversibly after a linear expansion.
F0 = µ ˙x ⇒ x =
F0
µ
t
1.1.2 Perfectly Elastic
On the contrary, a perfectly elastic model, retracts back completely to its
original self after expanding syn-chronically with the external pulse.
F0 = kx ⇒ x =
F0
k
1 Most of the diagrams given in this section are edited forms of graphs given in Ref-[3];
except for the fact that the axes are not to scale.
5

1.1.3 Maxwell liquid drop model
1.1.4 Linear viscoelastic solid model
Conclusions- It is interesting to note that the last spring-dashpot network
seems more plausible model for a living cell due to its capability for complete
recovery in spite of having ﬂuid-like properties.
Following the idea of complicating the system progressively, let us solve for
a similar cellular system in the next section.
6

2 A realistic cell
Let us now try to analyze diﬀerent types of macroscopic forces acting on a cell-
1. External force {still a step function}
2. A constant contractile force by the actomyosin cytoskeleton- fc
3. The cytoplasam is being modelled by the winner of last section-Linear
viscoelastic solid model; which is inherently elastic and viscous.
Figure 2: A realistic cell
The equation we would now like to solve is-
Fext = fc + kx + µ ˙x (1)
One can solve this in a piece-wise manner and then weave the solutions together.
2.0.5 Case-I [t < 1]
Let us solve for the time interval, where the external force on the cell is non-
zero.
So we have,
˙x =
(Fext − fc) − kx
µ
By integrating this, we get the solution for this patch with the given initial
conditions, as-
x(t) =
(Fext − fc)
k
(1 − e−( k
µ )t
) (2)
which can be written as-
x(t) = xmax(1 − e−t/τ
) (3)
where, the time constant for the system τ = µ
k and the maximum attainable
displacement xmax = (Fext−fc)
k (can easily be checked as the stable point for the
diﬀerential equation). But in the given time interval, the maximum it goes is
x(t = 1) ≡ x1.
7

2.0.6 Case-II [t > 1]
For the interval t > 1,Eq-(1) becomes-
0 = fc + kx + µ ˙x
One can quickly see, the stable point for this DE2
is xmin = −fc
k . It is easy to
reach this stable point by solving the DE starting from x1(2.0.5).
µ ˙x = −(fc + kx)
By integrating this equation with aforementioned initial condition,one can get
the solutions as-
x(t) = x1e∆t/τ
+ xmin(1 − e−∆t/τ
) (4)
Where, ∆t = t − 1.
Observations The solution in the second interval makes sense because the
memory of the initial condition fades away after a long time (indicated by the
first exponential term), as it exponentially progresses towards the new stable
point xmin (depicted by the second exponential term).
Figure 3: Putting xmin = ¯xf resolves the apparent discrepancy between the
nomenclature used in the text and the graphic.
Please note that if there was no contractile force, the final stable point would
have been simply zero. Because of the constant contractile force, the cell ends
up in a contracted state.
2DE- differential equation
8

3 Introducing Turnover
In our previous calculations, we assumed the contractile forces caused by the
actomyosin filaments to be time independent. However, it has been recognized
that- “ periodic shape changes in cells and tissues are indeed accompanied by
nearly anti-phasic cycles of cortical actomyosin density”.[1]
The new element in our model would be- a spatially homogeneous contrac-
tile element whose constituents are turning over.
The contractile element exchanges the force- producing motors (myosin mo-
tors in our case) of concentration c with reservoir. Please note that the τ used
here refers to the turnover time scale.
• Binding rate Kon = c0/τ
• Unbinding rate c Koff = c/τ
Therefore, net change in rate due to turnover becomes-
Kon − c Koff = −
1
τ
(c − c0)
But apart from the chemical kinetics, even the changing volume of the cell (in
response to external force) affects the concentration of myosin motors. Both
can be clubbed into one equation-
dc
dt
= −
1
τ
(c − c0) −
c
l
dl
dt
(5)
The second term ensures conservation of mass.
We also have the equation for length of the cell , fundamentally the same as
before, but the terms are a little more generalized this time.
Text + T(c) = K(l) + µ
dl
dt
(6)
Now lets look at the stability around the fixed point (l0, c0) by doing a linear
expansion.
µ
dl
dt
= Text + T0 − t1(c − c0) − K(l0) − k1(l − l0)
But right at the fixed point, we know currents vanish. So, we end up with-
Text + T0 = K0
Hence, the constants , hovering around, play absolutely no role.
µ
dl
dt
= −t1(c − c0) − k1(l − l0) (7)
9

Equations can be made dimensionless by-
˜c ≡
c
c0
, ˜t ≡
t
τ
and after making the following substitution- P1 ≡ µ
k1τ and P2 = ttc0
k1l0
Eq 7 becomes-
P1
d˜l
d˜t
= −P2(˜c − 1) − (˜l − 1) (8)
The operations render Eq5 beautiful too-
d˜l
d˜t
= −(˜c − 1) −
˜c
˜l
d˜l
d˜t
(9)
Note If P2 = 0 (one way to accomplish this is by making the contractile force
independent of motor concentration), the equations become decoupled and give
the trivial stable solutions i.e. (l0, c0); something we dealt extensively in section
2.
To gauge the stability of this (previously stable) fixed point, lets try lineariz-
ing the differential equations by substituting S = ˜c˜l. This helps us get rid of
the d˜l
d˜t
dependence in Eq 9 and makes it so much more beautiful!3
Re-framing the last statement in terms of the new convention- By substitut-
ing S = xy, we can get rid of the ˙y dependance.
y
dx
dt
+ x
dy
dt
= −(x − 1)y
⇒
d
dt
S = −xy + y = y − S
dS
dt
= y − S = fs(say) (10)
Similarly, from 8, we have-
a
dy
dt
= −b(x − 1) − (y − 1)
dy
dt
= −
a
b
(
S
y
− 1) −
1
a
(y − 1) = fy(say) (11)
To check the stability of the fixed point (which is the same as before i.e.
S = y and y = 1), we need to diagonalize the gradient matrix as per Appendix
Eq13.
3The overhead tildes bother me, so I chose to write in the classic (x, y) system, substituting
˜c −→ x, ˜l −→ y; P1 −→ a, P2 −→ b
10

The derivatives can be evaluated as-
∂fs
∂y P
= +1
∂fs
∂S P
= −1
∂fy
∂y P
=
1
a
(b − 1)
∂fy
∂S P
= −
b
a
Hence we need to solve,
⇒


−1 − λ 1
−
b
a
1
a
(b − 1) − λ

 = 0 (12)
This gives the quadratic equation-
λ2
+ λ(1 +
1 − b
a
) +
1
a
= 0
with solution (assuming a=1) being-
λ =
(b − 2) ± (b − 4)b
2
Conclusion Therefore we see that for different values of the parameters, the
solution either oscillates away from the fixed point or towards it in an expo-
nential fashion as observed in 3. Therefore the fixed point is no more trivially
stable after introducing turnover.
Figure 4: Length vs time
11

Figure 5: Concentration of motors vs time
Figure 6: Length vs. Concentration
12

A Appendix A
To gauge if a fixed point is stable or not, one can do the following-
1. Write down the rate change of variables in the following form-
dx
dt
= f1(x, y)
dy
dt
= f2(x, y)
2. Do a Taylor series expansion of the functions about the fixed point-
f1(x, y) =
∂f1
∂x P
x +
∂f1
∂y P
y
f2(x, y) =
∂f2
∂x P
x +
∂f2
∂y P
y
where P is the fixed point (x0, y0).
Note : The constant term at the beginning of the Taylor series is zero
by definition of a fixed point.
So now the rate equation has this matrix form.
d
dt
x
y
=




∂f1
∂x
∂f1
∂y
∂f2
∂x
∂f2
∂y




P
x
y
3. Diagonalize the gradient matrix so that we get the eigenvalues as the rate
along the eigen vectors.




∂f1
∂x
∂f1
∂y
∂f2
∂x
∂f2
∂y




P
x
y
= λ
x
y
=
λ 0
0 λ
x
y
⇒




∂f1
∂x P
− λ
∂f1
∂y P
∂f2
∂x P
∂f2
∂y P
− λ



 = 0 (13)
4. Solving for λ makes the solutions even more intuitive by letting us know
its time evolution.
d
dt
x
y
= λ
x
y
As these first order differential equations have exponential solutions, for real
eigenvalues, the solution will either exponentially stabilize to the fixed point, or
exponentially run away from it; for complex eigenvalues, it does the same with
an oscillatory behaviour.
13

References
[1] K. Dierkies, A. Sumi, J. Salon, G. Salbreux Spontaneous Oscillations of
Elastic Contractile Materials with Turnover PRL 113 , 148102, 2014.
[2] S. Banerjee, M. Christina Marchetti Contractile Stresses in Cohesive Layers
on Finite-thickness substrates. PRL 109, 108101 , 2012.
[3] C. T. Lim, E. H. Zhou, S. T. Quek Mechanical models for living cells- A
review. Journal of Biomechanics 39(2006)195-216, 2006.
[4] P. A. DiMilla, K. Barbee, D. A. Lauﬀenburger Mathematical model for the
eﬀects of adhesion and mechanics on cell migration speed. Biophysics J.
Volume 60, 15-37 , July 1991.
Acknowledgement I thank Prof. Christina Marchetti and Prof. Jeniifer for
guiding me through in spite of their hectic schedules. Most importantly, I extend
my gratitude to Scott, for it was his novel idea of making this project-work an
integral part of our coursework, that gave me the opportunity to delve deeper
into this beautiful territory and get hands-on with Mathematica after a long
time.
14

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