Review of “Emergent
Morphogenesis: elastic mechanics of
a self-deforming tissue”.
COURTESY: Pharyngula Blog, How to build a tadpole.
Davidson et.al Journal of
Biomechanics, 43, 63–70 (2010).
Bradly Alicea
http://www.msu.edu/~aliceabr
Why is this paper relevant?
Cell Biology
Development
Q: how are tissues self-assembled (from a
blastula to an embryo)?
Q: how do cells collectively form
structures in development/regeneration?
Emergent
Morphogenesis
Tissue
Engineering
Mathematics,
Physics,
Computer Science
Biomechanics
Q: how do physical processes affect the
nature and composition of cell populations?
Can we apply what we learn to:
* regenerative medicine (growth of tissues
on a scaffold)?
* selective phenotypic respecification,
selective differentiation?
Main Hypothesis
H: developmental processes can be modeled
and better understood using computational
geometry.
Gastrulation = geometric transformation.
* from sphere (A, left) to torus (A, middle)
to tube (A, right).
Gastrulation + elongation = deformation of
sphere along anterior-posterior axis (B).
Convergent extension = cell rearrangement
(C).
a) tissue deforms along anterior-posterior
axis.
b) oval with dot pushes itself in between the
other two ovals.
Previous Approaches to
Morphogenesis
D‟Arcy Thompson (1900): “On Growth and Form”
* mathematical rules govern developmental program
(relative position of cells in a structure).
* example: geometrical transformations determine species differences.
Alan Turing (1952): diffusion-based model.
* molecular interactions govern pattern-formation
(chemical diffusion).
Nature, 406,
131 (2000).
* example: striping in Drosophila embryo.
Modern approaches (see Curr. Opin. Genetics and Development, 14(4), 399–406):
* diffusion-advection models, refinement of Turing model.
* example: diffusion-limited aggregation model.
Goals of Computational Modeling
Computational Model: abstraction/representation and approximation.
Abstraction/representation: represent only the most “important” attributes of a
phenomenon.
Excitable brain
tissues as neural
network.
CAVEAT: we may not know what the most important parts are a priori.
Approximation: behavior of model is close to behavior of natural phenomenon.
CAVEAT: is behavior similar (mimicked only in certain situations), or is internal
mechanism?
Definition of Terms
Viscoelasticity: materials that exhibit both viscous and elastic properties when
deformed.
Convergent extension (CE): part of gastrulation, moves cells from three germ layers
into prospective positions (convergence + extension).
* brings prospective dorsal tissues from a broad area
of the early embryo and organizes them into a compact
column that runs from head to tail.
Xenopus spina bifida due
to KO in gastrulation.
COURTESY: http://www.
nibb.ac.jp/annual_report/
2002/html/02ann203.html
Intercalation : a reversible or irreversible inclusion of a cell between two other cells.
* mediolateral cell intercalation = cells
intercalate between adjacent neighbors, driving
them apart along AP axis. Several rounds
required to elongate embryo.
* fibronectin and fibrils at tissue interfaces
provide cues, keep all three cells in same
geometric plane.
Position and Mechanics as
Information
Mechanotransduction: the sensing by a living entity of mechanical cues that
result in an electric or chemical set of responses.
a) Nervous system: moving arm through a liquid, swinging a baseball bat.
* difference in mechanical forces, representation in nervous system (brain, spinal cord) of this information
(time-differencing + feedback = intelligent behavior).
* mechanisms: proprioceptors, joint receptors,
nociceptors.
b) Single cells: cells move across a surface
of specific hardness (motility), experience
pressure cues.
* difference in mechanical forces, changes in gene
expression, motile behavior, and cellular memory.
* mechanisms: cytoskeleton, focal adhesions, kinesins
(ECM).
Science, 315, 370-373, (2007)
Molecular-mechanical processes
Molecular-mechanical
understanding:
processes
1) source of birth defects.
underlying
1
are
crucial
to
5
2
6
2) formation of tumors and progression
3
of cancer (see 1-7).
7
3) principles of tissue engineering.
morphogenesis
Courtesy: Nature
Cell Biology, 9(9),
1010-1015 (2007).
4
Roles of mechanics in development:
Carbon Nanotube
tissue scaffold
1) Multicellular integration. Mechanical integration (tissue level) coordinates force
production, material properties of tissues, tissue movements (morphogenesis).
2) Intracellular cell integration. Mechanical integration of intracellular force generation with
local environment, direct molecular-mechanical processes (cell behavior).
(3) Intracellular gene integration. Mechanical integration of the cell, local environment, and
gene regulatory networks (direct cell differentiation).
Molecular-mechanical processes
(con’t)
Interactions between cell mechanics and
molecular /cell biology (examples):
1) coordinated movements of epithelial cells
during morphogenesis.
* build grooves, elongate tissues, and enclose
the embryo.
2) forces exerted by bone deformation trigger
signalling pathways.
*cells secrete extracellular matrix, nucleation
sites for minerals.
3) stromal cells react to external mechanical
loads by generating counteracting forces.
* exogenous (external) stresses and endogenous
(internal) tension.
Intercalation: “rules” and models
Intercalation drives convergent extention using
four “rules”:
1) Planar polarity. Intercalating cell moves in
mediolateral direction, separates neighboring cells
along anterior–posterior axis (A).
2) Remain in-the-plane. Intercalating cell stays in
same plane as the two neighbors (B).
3) Irreversibility. Intercalating cell does not reverse
direction and pop back out of plane (C).
4) Cell shape constraint. Intercalating cell and
neighboring cells maintain their shapes, do not reorganize within the same volume (D).
Intercalation: “rules” and models
In silico models simulate cellular tissues via cell–cell
adhesion, cell protrusive or traction forces, and cell
rearrangement.:
* make sense of the complex cell intercalation movements
Vertex
Cellular
Potts
FE
* account for the rates of cell rearrangement and cell shape changes that
have been quantified during Xenopus and zebrafish convergent
extention.
Model parameters based on empirical observations:
1) increase in mediolateral cell elongation length: width ratio
Xenopus
Zebrafish
Beginning of gastrulation
1.5
1.5
Beginning of neurulation
2.2
> 3.0
2) incidence of cell–cell neighbor change (does a cell change neighbors
during intercalation).
3) rate of cell neighbor change (probability that a cell contacts a new
neighbor during intercalation).
#1: Vertex Model
“Agent-based” model: autonomous agents use a ruleset to interact with each other and
form emergent patterns.
Spring
Autonomous Agent
Driven by stochastic processes (intrinsic randomness).
H: agent-based models can replace equation-based (deterministic) models. Emergence
= more than the sum of all parts (super-additive).
Vertex model: 2-D cell array, shared
boundaries between cells.
* boundary of each cell = Newtonian spring.
* stiffness, resting length parameters, uniform for
each cell.
#2: Cellular Potts Model
Cellular Potts models (CP, CPM): Agent-based model, simulate foam, biological
tissues (especially cell sorting – Nat. Rev. Genetics 10, 517-530, 2009), fluid flow.
* an array of “generalized” units: may be a single soap bubble, an entire biological cell, part of
a biological cell, or even a region of fluid.
CPM evolves by updating array
serially based on probabilistic rules.
* extended 2D model of convergent extension,
discrete packets of cytoplasm delimit contents
of each unit.
* cell sorting driven by adhesion/contractility.
Cellular Potts model for convergent extension:
* individual units, contiguous block within grid.
* cell-cell interactions governed by energy function
(heuristic measurement).
Additional rules maintain cell size, retard
shape changes (stiffness, energy penalty):
Inhibited: changes from pre-defined standards.
Encouraged: restoration of previous properties.
#3: Finite Element (FE) Model
Finite Element (FE) Model: a tool commonly used in Mechanical Engineering.
* can be used to study the biomechanics of early development.
* cellular representation: divide each cell into a mesh
(tiling of triangles, etc). Mesh discretizes continuous
surface.
Meshed cell
Use an energy function and deformations of the
structure to minimize potential energy.
* entire tissue structure is deformed in FE models
(global effect), energy minimized locally.
Autonomous Agent
FE Eye
FE Lung
Convergence and extension represented in two ways:
FE Head/
Neck
1) contractile rods representing traction forces along mediolateral axis can occur at random
locations, triggered using stochastic mechanism.
2) algorithms track cell-cell boundaries and cell rearrangement within the FE mesh.
Production of Four Rules in Vertex,
CP, and FE models
1) each model type modeled a different aspect of convergent extension (tissue
self-organization), but all established mediolateral cell intercalation:
a) Vertex model = elastically coupled cells.
b) CP model = differential adhesion.
c) FE model = mediolaterally-directed traction forces.
b)
COURTESY:
Nature Cell Biology,
5, 948 - 949 (2003)
c) centripedal traction
forces, mesenchymal
stem cell.
2) each model incorporated an approximation of cell stiffness.
Vertex (V)
Cellular Potts (CP)
Finite Element (FE)
Planar Polarity
YES
YES
NO
Remain in-plane
NO
NO
NO
Irreversibility
YES
NO
NO
Shape-Size Constraints
NO
YES
NO
Production of Four Rules in Vertex,
CP, and FE models (con’t)
Cell and tissue mechanics play an integral role in the emergence of convergent
extension.
* in particular, cell and tissue stiffness are critical for correct convergent extension.
Planar polarity: present in V and CP (but not FE) models, an outcome of local
feedback without initial bias.
* lack of bias is the criterion for convergent extension to be an emergent property.
Remain-in-plane: behavior is not reproduced by any model.
Irreversibility: reproduced only in V as a consequence of contact inhibition.
Cell shape and size constraints: not imposed in V nor FE models. May be a crude
approximation of shear and bulk stiffness.
“Wedge” Model
Simple equation rather than agent-based model (describe intercalation):
Ftraction > Felastic resistance (μs + μs cosα + sinα)
Traction of intercalating cell
must always be greater than
geometric constraints.
Ftraction (traction forces) overcomes friction (μ) and Felastic resistance (elastic resistance)
of neighboring cells
Length (L) and width (W) of intercalating cell determine intercalation angle (α),
where α = arctan (W/L).
Wedge phenomenon:
* intercalating cell acts as
wedge moving in between
two stationary cells.
* shape of cell determines success or failure of individual cell, how it fits into the
emerging monolayer, tissue structure.
“Wedge” Model (con’t)
A simple, semi-quantitative
alternative to the three prevailing
in silico approaches:
* forces required for intercalation
reduced for a more elongate cell
regardless of friction (B).
* if cell and environment
modeled as viscoelastic, cells
progressively elongate as they
intercalate.
* as the intercalating cell
„wedges‟ between its neighbors,
compressed
along
with
intercalating cell (C).
“Wedge” Model (con’t)
Behavior of artificial embryo in Wedge model:
* minimal amounts of resistance to elongation at anterior and posterior ends of (i.e.
boundary conditions) compress the cell in the AP direction.
* strain in response to wedging, resistance of the material to volume (e.g. Poisson
ratio), and compression along AP direction -- elongates cell in ML direction.
* these conclusions do not rely on specific shape of wedge but are general
properties of wedge model.
Behavior of the wedge model suggests feedback between intercalation, a reduction
in forces required for intercalation, and cell elongation occur in natural systems.
“Wedge” Model: broader
implications
Wedge model = simple machine that reproduce mediolateral cell elongation, take
into account cell shape and mechanical properties such (tissue stiffness, boundary
conditions).
* reveal universal mechanical principles that allow the broader fields of cell and developmental biology to
understand the complexities of morphogenesis.
Successful simulated models of CE:
* must produce correct shapes under conditions of stiffness, force production observed in real embryos.
Predictions of three models suggest a series of experimental tests:
1) do notochord cells elongate autonomously when dissociated or transplanted to other sites, or require
surrounding tissues (prediction of FE and vertex models)?
2) does cell elongation vary between embryo, free explants, or explants cultured on glass?
3) does the length-to-width ratio of cells match FE and Vertex models with their required boundary
conditions?
Conclusions and Applications
Future CE models should simulate the conditions in a more transparent,
biologically relevant manner.
* integrate signaling networks (e.g. non-canonical Wnt pathway).
Complex mechanisms that self-organize planar polarity without boundary
conditions, remain-in- plane, irreversibility, and cell shape constraint, future
models will need to:
1) autonomously generate subcellular polarization that can direct mediolateral cell protrusions.
2) represent multiple cell layers or even full 3D cells.
3) autonomously generate persistent cell intercalation behaviors.
4) represent more realistic material properties of both intercalating cells and the tissues they form.
Also need flexibility to include specialized adaptations for different organisms
(mouse, chick, and zebrafish) and organ formation.
Conclusions and Applications
Microenvironments for stem cell niche:
Biophysical, geometric aspects of ECM,
affect retained self-renewal capacity
(Keung et.al, WIREs Sys. Bio. Medicine,
2, 49-64, 2010).
Tissue engineering at small scales:
* how can we have greater control over
stochastic mechanisms?
* how can we produce features of a
specific, characteristic size?
* can we use simulation, experiment to
predict by-products of emergent
properties?
* effect of variables such as
dimensionality, linear elastic modulus,
stiffness?