The effects of time delays in models of cellular pattern formation

Nick Monk, University of SheffieldSiren Veflingstad & Erik Plahte,

Norwegian University of Life Sciences

Multicellular development: pattern formation

• During the development of multicellular organisms, differential

fates must be assigned to cells that have equivalent potential.

• Cell states must be flexible (not specified by lineage), but also

stable and heritable. This can be achieved by modulating the

levels of expression of genes within cells.

• The establishment of appropriate patterns of cell fate depends on

intercellular signalling. Cells constantly exchange information on

their current state with their neighbours (either locally, or at a

distance).

Cell fate assignment can be very rapid

Don Kane & Rolf Karlstrom, 1996

Lateral inhibition

• Many fate decisions are binary. There is usually a default fate

that a cell will adopt autonomously (in the absence of

interactions). This is termed the primary fate.

The other fate is the secondary fate.

• Lateral inhibition is a type of intercellular interaction by which a

cell adopting the primary fate can inhibit its immediate neighbours

from doing the same (and so they adopt the secondary fate).

• Lateral inhibition is used in many different settings in

development, and uses a conserved signalling pathway.

Renaud & Simpson, Dev. Biol. 240, 361–376 (2001).

Delta, Notch,…

Stochastic fate assignment

Loss of key genes involved in the Delta-Notch signalling pathway leads to over-assignment of bristles (1º fate)

Pattern is achieved rapidly (ca. 2 hours)

Dl and N are expressed uniformly during assignment

Lateral inhibition: Drosophila bristle spacing

Genetic data suggest:

• Dl activates N on neighbouring cells (juxtacrine signalling)

• N activity represses Dl “activity” (within the same cell)

• N activity determines cell fate (via regulated transcription)

• Low N activity 1° fate

• High N activity 2° fate

• Dl–N signalling amplifies differences in activity between cells (stochastic or imposed)

Competition: Delta-Notch signalling

Delta-Notch signalling: ODE model

• Ni and Di are Notch and Delta activities in cell i

Di = (Di–1+ Di+1)/2 = average of D in cells neighbouring cell i

• fS and fR are functions representing signalling and intracellular regulation

• fB is monotonic increasing and fR is monotonic decreasing (:[0,∞] [0,1])

N and D are 1st order degradation rates, pN and pD are maximal production rates

Collier, Monk, Maini, Lewis. J. theor. Biol. 183, 429–446 (1996).

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dN i /dt = −μN N i + pN fS Di( )

dDi /dt = −μDDi + pD fR N i( )

Consider a line of cells, labeled by a single index i

Collier et al., J. theor. Biol. 183, 429–446 (1996).

Dynamics of lateral inhibition

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D and N often approach the (unstable) homogeneous steady state before diverging to the patterned steady state.

This can result in transient homogenisation.

Initial approach to homogeneous steady state

The Drosophila neurogenic network

ODE Model: Meir, von Dassow, Munro, Odell. Curr. Biol. 12, 778–786 (2002).

• There is an irreducible delay (typically 10–20 min) from initiation of a transcript to appearance of functional mRNA in the cytoplasm.

• The delay can be much longer (16hrs for human dystrophin).

• Translation delays are shorter, and can be (formally) incorporated into the transcriptional delay (by ‘lagging’ a variable).

• Delay equations should be used to model transcription when the delays are of the same order as the system dynamics. Delays are a reality, not an hypothesis.

Eukaryotic transcription: time delays

functional mRNA(OUTPUT)

(INPUT)

Monk, N.A.M. Curr. Biol. 13, 1409–1413 (2003).

D2N1

D1 N2

To account for the three transcriptional steps in the neurogenic network, a delay (of around an hour) should be incorporated in the competition model (Delta alone takes ~20 min to transcribe).

Deal first with the simple model to assess the effect of the delay.

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dN i /dt = −μN N i + pN fS Di( )

dDi /dt = −μDDi + pD fR N i(t − τ )[ ]

(or distributed delay equivalents)

A delayed competition model

= 3.5

The delay model has oscillatory transients

Veflingstad, Plahte, Monk. Physica D 207, 254–271 (2005).

For general initial conditions, the time taken to pattern grows rapidly with the delay.

Spatial pattern grows slowly

= 3.5

“best case” scenario: growth of pattern from homogeneous steady state (hss). One cell on each side of hss.

If non-delayed model takes 2 hours to pattern, the model with a 1 hour delay takes ca. 14 hours.

The neurogenic model is a poor patterner

4 8 16 2012

• So far, have driven patterning by initial conditions.

• Instead, initiate pattern by driving the system through a bifurcation by parameter-modulation (and noise).

• No oscillatory transients, but slow.

Importance of “initial conditions”

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dD1 /dt = −μ(t)D1 + f D2(t − τ )[ ] + ξ1(t)

dD2 /dt = −μ(t)D2 + f D1(t − τ )[ ] + ξ 2(t)

0 10t (hours)

(t) (hours–1)

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Interacting loops (multiple delays) can lead to more complex oscillatory behaviour (e.g. Lewis, J. Curr. Biol. 13, 1398–1408 (2003)). However, transcriptional delays always lead to significant delays in patterning.

More detail (probably) doesn’t help

Is it all bad news?

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Delayed Notch signalling is a powerful homogeniser. This is also important during development.

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• Random starting states:

• Spatially patterned (sinusoidal) starting states:

Transcriptional oscillations: somitogenesis

• Propagating oscillatory transcription of genes such as Fringe, Delta, Hes/Her.

• Period of oscillation:

• Zebrafish: 30min

• Chick: 90min

• Mouse: 120min

Propagating pulses of Notch activity

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• The somite oscillator depends on graded gene expression along the body axis.

• As an example, incorporate this in the model as a gradient in the degradation rate of Delta activity:

Rostral Caudal

1. Time delays (due to transcription/translation) have significant effects in models of pattern formation.

2. Transients and time-scales are critical. The N Dl interaction is unlikely to be mediated by transcription (during competition). Is there a post-translational short-circuit (c.f. C. elegans)?

3. Delayed signalling may underlie more complex spatio-temporal modes of pattern formation, such as in somitogenesis.

Conclusions