7/28/2019 MacLennan FNANO 2013: Mathematical Principles of Morphogenesis Applied to Nanoscale Self- Assembly

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RESEARCH POSTER PRESENTATION DESIGN 2012

www.PosterPresentations.com

The Challenge: How can we coordinate

the behavior of millions of microscopic

agents to assemble complex,hierarchically structured macroscopic

systems?

Hypothesis: The morphogenetic processes

that operate in embryological

development can be applied to the self-

assembly of complex, hierarchical

systems.

OBJECTIVES

APROGRAMMINGLANGUAGE

FORMORPHOGENESIS

Assemble a segmented spine with a pair of segmented legs on each

spinal segment.

Control the number and length of spinal and leg segments. Control the position of the legs.

EXAMPLEPROBLEM

Caudal Morphogen:

Rapidly accumulates in tail

tissue Diffuses and degrades

Represents proximity to tail

tissue

Rostral Morphogen:

Accumulates in differentiated

segments (S> 0)

Diffuses and degrades

Represents proximity to

differentiated segments

CAUDALANDROSTRALMORPHOGENS

DIFFERENTIATIONOFIMAGINALDISKSImaginal disk tissue differentiates when:

anterior border morphogen in correct

range

posterior border morphogen in correct

range

segment density is sufficiently low

REFERENCES

J. Cooke, E.C. Zeeman (1976). A clock and wavefront model for control of

the number of repeated structures during animal morphogenesis, Journal

of Theoretical Biology58: 455476.

B.J. MacLennan (2010). Morphogenesis as a model for nano

communication,Nano Communication Networks Journal1: 199208.

B.J. MacLennan (2012). Molecular coordination of hierarchical self-

assembly, Nano Communication Networks Journal3: 116128.

B.J. MacLennan (2012). Embodied Computation: Applying the physics of

computation to artificial morphogenesis,Parallel Processing Letters 22:

124013.

I. Salazar-Ciudad, J. Jernvall, S. Newman (2003). Mechanisms of pattern

formation in development and evolution,Development130: 20272037.

MOREINFORMATION?

Email: MacLennan@utk.edu

Web: web.eecs.utk.edu/~mclennan [sic]

FIRSTSTEPSTOWARDLEGGROWTH

Change equations for describing discrete- or continuous-time

behavior:

Example definition of a diffusible substance:

Self-assembly of complex, hierarchically structured systems from

microscopic components will require artificial morphogenesis, inspired

by embryological development

This entails understanding the mathematical structure of morphogenetic

processes and applying it in artificial systems

Use of a PDE-based notation facilitates scaling to v ery large numbers of

components

As an example we have applied the clock-and-wavefront process to

simulated assembly of a complex object

Posterior Segment Border:

Segment tissue differentiates into

posterior border tissue when:

segmentation signal () passes through

caudal morphogen (C) concentration is

high

Anterior Segment Border:

Segment tissue differentiates into

anterior border tissue when:

segmentation signal () passes

through

rostral morphogen (R) concentration is

high

Adopt mathematical descriptions of natural morphogenetic processes to

artificial systems

Ensure processes will scale up to millions

or billions of agents by going to the

continuum limit (stochastic PDEs)

Nevertheless, maintain complementarity

between discrete and continuous models

Continuum mechanics of visco-elasticmaterials (soft matter)

Mimic or replace the fundamental morphogenetic processes described

by Salazar-Ciudad, Jernvall, and Newman (2003)

Components are:

Both active and passive

Simple, local sensors (chemical, etc.)

Simple effectors

local action (motion, shape, adhesion) signal production (chemical, etc.)

Simple regulatory circuits (need not be electrical)

Ambient energy and/or fuel

Self-reproducing or not

T= density of tissue in terminal (tailbud) state

u = direction of motion

r= rate of movement or growth

M= density of undifferentiated tissue = length of tailbud

Inverse quorum sensing: detect when density of neighbors is below a

threshold

Implemented by morphogen diffusing from segment tissue Modeled by convolution with Gaussian

kernel determined by diffusion parameters

Imaginal disk tissue differentiates to be in T

(terminal) state

These cell orient outward (i.e., grad S)

Begin to move and produ ce undifferentiated legtissue (ready for clock-and-wavefront)

The figure shows the formation of two segments

of the first leg pair

Residual morphogens interfere with correct

formation of the second pair.

More work to be done!

Anterior/Posterior Position:

Anterior and posterior border

tissues emit anterior (a) and

posterior (p) morphogens, which

diffuse and degrade

Establish opposing gradients by

which position can be determined

Tissue differentiates into segment tissue when:

segmentation signal () passes

through

sufficiently far from tail

(C< threshold)

sufficiently far from previous segments (R < threshold)

The tissue is an active medium

Clock signal causes a patch of tail tissue to fire: emit a pulse of

(segmentation morphogen)

It diffuses and degrades

Sufficiently high stimulates nearby tissue to fire

But after tissue fires, it enters a refractory period (determined by a

variable )

Ensures unidirectional propagation

Vertebrae: humans have 33, chickens 35, mice 65, corn snake 315

characteristic of species

How does developing

embryo count them?

Segments also govern

development of organs

Clock-and-wavefront model

of Cooke & Zeeman (1976),recently confirmed (2008)

Depends on clock, excitable

medium (cell-to-cell signaling), and diffusion

-DR = DRr2RR/R + RS(1R)

= > ^ < M

-D + = [G > G ^K > K]T

-D + = + Dr2 /

-D = /

EXTERIORSURFACEDETECTION

GOALOFSPINALMORPHOGENESIS

APPROACH

MICROROBOTS,CELLS&MACROMOLECULES

GROWTHOFUNDIFFERENTIATEDTISSUE

LOCATIONOFIMAGINALDISKS

SEGMENTPOLARIZATION

SEGMENTDIFFERENTIATION

WAVEPROPAGATION

CLOCK-AND-WAVEFRONTPROCESS

CONCLUSIONS

DepartmentofElectricalEngineering&ComputerScience,UniversityofTennessee,Knoxville

BruceJ.MacLennan,PhD

Mathema/calPrinciplesofMorphogenesisAppliedtoNanoscaleSelf-Assembly

-DC= DCr2C C/C+ CT(1 C)

- S + = > lwb ^ C < Cupb ^ R < Rupb

-DS + = SS(1 S)

-DA + = [AupbRupb > R > AlwbRupb

^ > lwb]

-DA + = ASA(1A)A/A

-DP + = [PupbCupb > C > PlwbCupb

^ > lwb]

-DP + = PSP(1P) P/P

- I = [aupb > a > alwb

^ pupb > p > plwb

^ S < Supb]

S(1 I)

E = [ S < Supb]

-Da = [A > A]aS(1 a) + Dar2a a/a

-Dp = [P > P]pS(1p) + Dpr2pp/p

r = [G > G]r0-DT = (rTu) = r(u T + T u)

- M = rT/

substance morphogen:

scalar field concentrationvector fields:

j flux drift vector

order-2 field diusion tensor

behavior:

j = (

T

)/2 flux-D = j change in conc.

-DX= F(X,Y)

Controlled sequence of differentiated

segments

Anterior and posterior regions of

segments further differentiated