Acknowledgement

Pressure of a cell and tension of an edge
A top view ofan epithelial sheet	

A side view ofan epithelial sheet	

Myosin,Cortical actin cable
10 µm	

1 min interval	
 Replay: 15 fps	Balance of forcesat each vertex

Pressure

T1 P1
P3

Cadherin	Tension	Cut an edge!

+	T2

T3 P2

Cytoplasm,
microtuble	

Myosin, Corticalactin, Cadherin	

5 µm	

Questions

Morphology

Mechanical force

1. How does mechanical force that drives tissue
deformation emerge in light of the molecularproperties of the constituent cells?
2. How does the generated forces regulate thecorrect patterning of a tissue?
Tissue morphogenesis
Cellular behaviors
Molecular dynamics
Germband elongation in
the Drosophila embryo
Bertret et al. (2004)

Cell Rearrangement
Lecuit lab, Zallen lab etc. 	

Myosin Cadherin

Measuring dynamics of forces in vivo is very difficult
A relative value of tension
measured by laser cutting	0 sec	

15 sec	

30 sec	45 sec	Destroy corticalactin network	

30 sec	45 sec	“invasive”	

Verticesmove fasterLarger
tension	

Balance of forcesat each vertex

No appropriate method
to directly andrepeatedlymeasure forces
in a multi-cellular tissue

Pressure

T1 P1
P3Tension	

Cut an edge!

+	T2

T3 P2

Cytoplasm,
microtuble	

Myosin, Corticalactin, Cadherin	

5 µm	

Outline
1. Our new method to estimate forces
2. Forces in the Drosophila wing
3. How forces regulate epithelial patternformation

P	A	

P	A	

D	P	

D	P	

Outline of our new method to estimate forces	
positionof vertex	

v1	v2	

v1(x1, y1),v2(x2, y2), 	

balance	
  of	  forces	
  at	  each	  vertex	
  α 1x T1 + α x2T2 + α x3T3 + β 1x P1 + β x2 P2 + β x3 P3 = γX˙ 0
α 1y T1 + α y2T2 + α y3T3 + β 1y P1 + β y2 P2 + β y3 P3 = γY˙0x −x
T = 1  0 × T1
x1 − x 0x1€€drag	x1p = (T1 ,TM ,P1,,PN ) V = x˙

Output:	  	  “relative” values of€forces	€“	  minimize	  	  	  	  F =	
  |Αp-γV|2	  	  +	  	  µ|Βp -f |2	  	  ”	  	  y − y1
P = 2× P12
Ap =€ γV

pressure & tension	

conditions< unknowns	
Inverse	
  problem	
  High	High	Pressure	Tension	

Low	Low	µ is determinedby Bayesianstatistics

Expectedfeaturesof system	

=of cells	
Input: image	

Positivetension	

Indefiniteness and its physical interpretation
R1- N+R cells, v+R vertices, and M+2R edges- If all vertices are 3-way junction, M=(3/2)v- Note Euler’s theorem v-M+N=12 3
1 2 3

Conditions: (num. of vetex)×2= 2(v+R)
Unknowns: (num. of cell + edge) = M+N+3RN

	  1	  indefiniteness byhydrostatic pressure	Apiso = 0,piso = (0,,0,1,,1)

	
  R	
  indefiniteness byboundary condition€R+1 indefiniteness 	Estimate relative valuesof forces	

Pi = cpi + phs,T j = ct j
Becomes less significant
as N increases	

True and estimated forces in artificial data
Numerical	
  simula8on	
  : 	U = ∑U e (Ai ) + ∑U a (l j ) + ∑U c (Li )Cell:iContact: j

Cell:iArea
elasticity

Cell adhesion, CorticalContraction
elasticity

Calculate	
  true	  values	
  of	  forces	

係数を振る	

Es8ma8on	
  : 	

Estimated values	€PressureTension	
True values	※ The prior (positive tension) that we adopted yielded the best
results among priors tested (e.g. L2-norm, spatial smoothness）.

Advantages of our new method

Input: 	

image of cells	

Output:	  	   distribution of forces	 Single cell resolution
 Applicable to a tissue-scale Noninvasive No assumption of forms of the potential energy

Drosophila epithelial tissues (notum and wing)
Larva"

Wing disc of Drosophila 3rd instar larva"

Embryo"http://flybase.org/	
Anterior"

はね

翅原基

Ventral"

100 µm"Pupa"

Adult"

Wing at 15.5 hr APF (after puparium formation)"
Wing"
Anterior"

10µm	
50 µm"

Proximal"Application to the Drosophila pupal wing	Input: image	of cells	

get	
  posi8ons	
  of	
  ver8ces	

20 µm	

va(xa, ya),vb(xb, yb), 	
Ap = γVbalance	
  of	  forces	
  at	
  each	  vertex	
  pressure & tension 	drag	
p = (T1 ,TM ,P1,,PN ) 
V = X˙+	Bayesianformulation	

€values of forces	
Output:	
  “relative”	
  €A	

-0.04	
0	 0.02	
lowpressurehigh	0.6	
low

1.0	
tension

1.4	
high	P	D	P	The summary of results
1. Our new method to estimate forces2. Forces in the Drosophila wing
- stronger tension along the PD axis
- nature of tension: external force stretching the wing- ”force-generating” and “force-responding” properties of myosin

3. How forces regulate epithelial patternformation- hexagonal packing in the wing and notum- more strongly biased tensile force -> more hexagons?
- biased external force accelerates hexagonal pattern formation both insilico and in vivo

Anisotropic external force instructs directionalarrangement of cells for efficient hexagonal packing
Anisotropic stretch

Anisotropic tensile forces may avoid
the mismatch in orientations of small
group of cells by setting the uniqueaxis of cell rearrangement.	

Biased external force

Aligned!!	
Faster relaxation of the energy

Control and isotropic expansion

5	 7	 5	This mismatch in orientations of small
group of cells produces non-hexagonalcells. 	

Mismatch!!	

Slower relaxation of the energy

Mechanical basis of pattern formation:
feedback regulations that connects molecular,cellular, and tissue dynamics 	
TissueCell
Molecule

Stress field of a tissue
Hexagonal packing
Junctional remodeling
MyosinLowTensionHigh	

100µm