The cellular cytoskeleton is essential in proper cell function as well as in organism development. These polymers provide the elaborate roads along which most intracellular protein transport occurs. I will discuss several examples where mathematical modeling, analysis, and simulation tools help us study and understand the interactions between these filaments roads and motor proteins in cells. In neurons, neurofilaments navigate axons and their constrictions to maintain a healthy speed of neuronal communication. We develop stochastic models that may provide insights into transport mechanisms through axonal constrictions. In the reproductive system of the worm C. elegans, we use agent - based modeling to study how myosin motors interact with actin filaments to maintain contractile rings that allow passage and nutrient transport for developing egg cells. In addition, we have recently become interested in using topological data analysis tools to assess maintenance and establishment of these ring structures.

1. MATHEMATICAL
BIOSCIENCES INSTITUTE
• To foster innovation in the application of mathematical,
statistical, and computational methods in the resolution of
significant problems in the biosciences;
• To foster the development of new areas in the mathematical
sciences motivated by important questions in the biosciences;
• To engage mathematical and biological scientists in these
pursuits; and
• To expand the community of scholars in mathematical
biosciences through education, training, and support of students
and researchers.

2. MBI EMPHASIS SEMESTER ON ANALYZING
MACRO AND MICRO POPULATION MODELS
Current Topics Workshop: Collective behavior and
emergent phenomena in biology (Sept 10-12, 2018)
Modeling for Family Trees in Disease Studies (Sept 17-
19)
Math and Microbiome (Oct 10-12)
Modeling and analysis of dynamic social networks (Nov
5-9)

3. MODELING THE CYTOSKELETON ROADS
IN INTRACELLULAR TRANSPORT
Modern Mathematics Workshop
October 11th 2018
Veronica Ciocanel
Mathematical Biosciences Institute at OSU
https://valelab.ucsf.edu/motility/

4. THE BROAD QUESTIONS
For cells to function properly, protein molecules
and structures have to move around and organize
into complex patterns.
How do they move/organize in the precise time
and spatial scales observed inside cells?
What are the mechanisms of transport and
organization?
4
Video source: Vale Lab at https://valelab.ucsf.edu/motility/

5. Cellular
cytoskeleton
Example: elaborate cellular
roads in neurons
Fletcher, D. A. & Mullins, R. D. Cell mechanics and the cytoskeleton. Nature 463, 485-492 (2010)
5

6. Fletcher, D. A. & Mullins, R. D. Cell mechanics and the cytoskeleton. Nature 463, 485-492 (2010)
6
Cellular
cytoskeleton
Microtubules
Björn Sandstede and
Kim Mowry, Brown University

7. Microtubules
Fletcher, D. A. & Mullins, R. D. Cell mechanics and the cytoskeleton. Nature 463, 485-492 (2010)
7
Cellular
cytoskeleton
Intermediate filaments
Tony Brown, OSU and
Peter Jung, Ohio University

8. Microtubules
Fletcher, D. A. & Mullins, R. D. Cell mechanics and the cytoskeleton. Nature 463, 485-492 (2010)
8
Cellular
cytoskeleton
Intermediate
filaments
Actin filaments
Adriana Dawes, OSU

9. 9
1. mRNA transport
along microtubules
Bjorn Sandstede Kim Mowry

10. THE QUESTION
In early development, many organisms have
distinct spatial features and patterns.
Asymmetries are achieved through
asymmetric accumulation of messenger RNAs
(mRNAs).
A key mechanism is active transport on
microtubules.
Gagnon et al., 2013 (Mowry Lab)

11. MODELING
11
Account for dynamics such as:
Diffusion of free mRNA particles
Active transport of motor-mRNA complexes
Unbinding from / binding to microtubules
Diffusion and transport states

12. 12
1C., Kreiling, Gagnon, Mowry and Sandstede, 2017; 2Powrie, C. et al, 2016.
PREVIOUS WORK
Transport model PDEs
Dynamical systems analysis for
large time

13. 13
MICROTUBULES:
THEORETICAL INSIGHTS
2-state model with diffusion (off MTs) and active transport (on MTs).
Assumptions: small diffusion, very fast transition rates, small MT density;
thus, quasi-steady-state analysis.
Not the case for our parameter estimates.
Hawkins, 2009; Bressloff and Xu, 2015:

14. Theorem
C. and Sandstede, in review
14

15. MICROTUBULES:
SIMULATION INSIGHTS
15
Microtubules: Messitt et al., 2008
In Xenopus oocytes, microtubules are random with a radial bias.
C., Jeschonek, Sandstede, and Mowry in review

16. 16
2. Neurofilament transport in
axons
Tony Brown, OSU Peter Jung, Ohio University

17. MYELINATION AND AXONAL
TRANSPORT
Synapse
Axon
Cell body
A motor neuron
Axonal transport
Saltatory nerve conduction
The nervous system

18. AXONS ARE CONSTRICTED AT
NODES OF RANVIER
Thus nodes are potential bottlenecks for axonal transport!

19. Mouse saphenous nerve,
freeze-substitution
Okamura & Tsukita, 1986
How do
neurofilaments
navigate these
nodal
bottlenecks?
Hypothesis:
Nodal acceleration can
be explained by a local
increase in the average
proximity of
neurofilaments to their
microtubule tracks
FEWER NEUROFILAMENTS IN NODES

20. 20
STOCHASTIC MODELING
Switching between short- and
long-term pauses (experimental
and computational support).
Model the neurofilaments in one
dimension along the axon
Keep track of NF content
throughout the axon
1 µm
Axon
NFs
vol(n): total NFs at bin n
occ(n): on-track NFs at bin n
n

21. 21
STOCHASTIC MODELING
No available MT tracks for
NFs at a location: set velocity
and on-rate to 0.
Multiple available tracks:
velocity and on-rate decrease
with the number of on-track
NFs.
STOP
GO
MT

22. INTERNODE SIMULATIONS
23

23. 24
Can we reproduce NF
densities at nodes through a
change in the on-rate
(availability to microtubules)?
NODE SIMULATIONS

24. 25
Can we reproduce NF
densities at nodes through a
change in the on-rate
(availability to microtubules)?
NODE SIMULATIONS

25. 26
Average NF length influences the sharpness of the node flanking
regions.
Can we back out NF length information from these simulations?
MODEL EXPLORATION
5 µm 20 µm

26. 27
The model predicts fluctuations in NF content throughout
the axon. How uniform is this content?
Model assumes one-dimensional dynamics. How do
predictions change when introducing the axon width?
Abnormal NF accumulations are reported close to nodes of
Ranvier in neurodegenerative diseases. What mechanism
defects may be responsible for NF accumulations?
FUTURE WORK

27. 28
3. Actin-myosin ring
channels and interactions
Adriana Dawes, OSU
Math and Molecular Genetics

28. 29
Why study ring channels?
q Cells use many types of channels to communicate, each with different
functions.
q Ring channels play critical roles in oogenesis, wound healing, and cytokinesis.
q Often, they maintain precise diameters over a considerable time scale.
q Non-muscle myosin motors are often “in charge” of creating appropriate
constriction of the ring.
http://rusty.fhl.washington.edu/celldynamics/events/workshops/archive/2003/cytomod_abstracts/BBement/index.html

29. THE QUESTION
q Two myosin motors believed to function similarly are involved in ring
channel dynamics.
q RNAi experiments for these motors show that they may antagonize
each other in the ring channels with respect to cellularization.

30. GOALS
q Some preliminary modeling suggests both
tangential and normal activity of the motors.
q Test this idea using agent-based models for
actin filaments and motors.
q Identify differences in the two types of
motors.
q Rigorous analysis of data from experiments
and from simulations.

31. AGENT-BASED MODELING AND
SIMULATION
q Accounts for dynamics and molecular
transport of chemical species.
q Diffusion and active transport are
modeled as stochastic jumps between
compartments.
q Is based on energy minimization.
Medyan (Papoian Lab)
Popov, Komianos, and Papoian, 2016

32. SIMULATIONS
Standard With formin nucleation
actin
filament
myosin
motor
cross-
linker
formin

33. HOW TO SYSTEMATICALLY
DETECT RING STRUCTURE?
Small on-rate Large on-rate
actin
filament
myosin
motor
cross-
linker

34. HOW TO SYSTEMATICALLY
DETECT RING STRUCTURE?

35. 36
TOPOLOGICAL DATA ANALYSIS
Topaz, Zieglemeier, and Halverson, 2015

36. 37
TOPOLOGICAL DATA ANALYSIS
Topaz, Zieglemeier, and Halverson, 2015

37. 42
FUTURE WORK
q Refine simulations and implement/model:
q Two types of filaments
q Two types of motors with different parameters
q Application of forces to the domain and comparison to in
vitro motor results
q Stabilization of actin dynamics and comparison to in vitro
motor results.

38. Thank you for your attention!
43
If you have any further questions/suggestions, get in
touch at ciocanel.1@mbi.osu.edu.
Twitter: @v_ciocanel.