Martin Howard

Dept of Systems Biology John Innes Centre, UK

Modelling Noisy Concentration Gradients in Developmental Biology

Position determination in biology• How to measure position in biological systems? • One solution: use gradients of protein concentration, created, for example, by localised protein production but global degradation:

• Absolute position: if local concentration is above threshold: switch on downstream signal

Absolute position: a (very) simple gradient model• Localised activation at x=0 at rate J• Diffusion constant D• Uniform deactivation at rate μ• Length L in x-direction

at steady-state and assuming

2 ( )D J xt

( ) expJ xxD

• Very simple model, but potentially still biologically relevant!

• Morphogens Bicoid, Dpp, Wingless in Drosophila all have exponential profiles

LD /

Example: Bicoid!• Intensively studied example: Bicoid protein in fruit fly Drosophila melanogaster

Houchmandzadeh et al. Nature (2001)

• Gradient well fitted by simple exponential

• Bicoid drives very precise gene expression of hunchback!

Noisy gradients Noise can be due to external or internal variation

External fluctuations givecell-to-cell or embryo-to-embryo variability

Internal fluctuationsaffect accuracy withina single cell/embryo

• Internal fluctuations give limit to precision of positional information: can we calculate this?!

Imprecision in positional information• Consider a volume (Δx)d centred at x• Number of particles within volume is n(x)• Diffusion, decay and production each give Poisson

statistics for particle number. Since the system is linear, overall fluctuations must also be Poisson:

22 ( ,0) ( ,0) ( ,0)n x n x n x

22 2 ( ,0)( ,0) ( ,0) ( ,0)

( )dx

x x xx

• Now convert fluctuations in density to error in

0 exp( ) 2

Td

xDwJ x

positional information, given by a width:

Tostevin, ten Wolde, Howard PLoS Comput Biol (2007)

See also Gregor et al, Cell (2007))()0,(

0T

T

xxw

Imprecision will be large!• Identify (Δx) as the size of the detector• In developmental biology, if morphogen is a transcription factor:

detector will be binding target in regulatory DNA• Both cases: appropriate scale ~5nm• Both cases: protein copies sparsely distributed large error

Reducing the imprecision: time-averaging

• Integrating for time we can make Nind independent measurements

• ind~(Δx)2/D is the typical time required for diffusion to refresh the detector region

• Expect concentration fluctuations to go as

and width as:

• Precision maximised for a particular choice λ=xT!

( ,0)( , ) xxN

2 exp( ) 2T

d

xw kJ x

Spatial analog of Berg-Purcell result (1977)

Data collapse for long averaging times:

SimulationsFor xT=2μm, w is minimised at λ=2μm

In d=2, w is independent of Δx (up to log correction)

• Reducing Δx reduces the number of particles being measured at each site, so increases fluctuations.

• But it also increases number of independent measurements in t.

In d=2, these effects cancel!

2 exp( ) 2T

d

xw kJ x

Role of detector size

Noisy gradients Noise can be due to external or internal variation

External fluctuations givecell-to-cell or embryo-to-embryo variability

Internal fluctuationsaffect accuracy withina single cell/embryo

• So now we understand internal fluctuations, but what about the effect of external fluctuations?

Combining External and Internal Fluctuations• Calculate imprecision W in positional information due to embryo-to-embryo fluctuations

• Focus on fluctuations δJ in injection rate J

• Doesn’t improve through time-averaging!

• Internal and external noise are statistically independent

• So total imprecision in positional information is given by a width ε

• Can the total imprecision be minimised?

JJW

22 Ww

Saunders & Howard, Phys Rev E (2009)

• Total width εlin given by

internal noise external noise

• Minimise width as a function of λ

• Use parameters inspired by the Bicoid gradient, with spatial averaging and 5 min time averaging

• Optimising kinetic parameters can have a substantial impact on the precision of the positional information!

Maximising precision

22/0

212 )/(

JJe

xNJDDk

Tx

spatlin

Saunders & Howard, Phys Rev E (2009)

What about other gradient shapes?

• So far assumed an exponential profile in agreement with data on Bicoid, Dpp, Wingless

• But could have other shapes

• Could these shapes give better positional information?

• Introduce two further representative shapes:

- power law generated by quadratic degradation model

- linear generated by source-sink model

• Decay via dimerisation process

• Can solve profile exactly to give

with and

• Asymptotically a power law for x»x0

• Quadratic decay profiles more robust to external fluctuations in J

• Question: why aren’t all morphogen profiles power laws?!!

• Could this be due to internal noise?

Quadratic degradation model

)(22 xJDt

20 )(

)(xxAx

DA 6

31

2

012

JDx

JJxWquad

30

Barkai et al. Dev. Cell (2003)

Statistics in the quadratic decay model• Perform similar calculation to before to compute internal and external noise

• But quadratic decay model is nonlinear; so what are the statistics of the internal noise?

• Still Poisson!

• Non-Poissoninan statistics due to nonlinear reactions are mixed away by diffusion as d=3 is above the upper critical dimension of dc=2

)(

)()()()(

22

xn

xnxnxnxf

Maximising precision in the quadratic decay model

220

30

400

222

9)(

2)/(

JJx

xxx

xNJDDk T

spatquad

Saunders & Howard, Phys Rev E (2009)

• Calculate total imprecision in positional information:

internal noise external noise

• Again precision can be maximised as a function of x0!

• Optimising kinetic paramaters to maximise positional information is a general feature of morphogen gradients

Source-sink model• Absorbing sink at x=L

• Biologically realistic if degrading enzymes are themselves localised

and

• Internal fluctuations again Poissonian

• Can calculate precision of positional information:

• So which of our three models performs best?!

)(2 xJDt 0Lx

220

202 )()()/(

J

JxLxLxNJDDk

TTspat

ss

Comparing the models• Which model is best depends on the averaging time!

• Short averaging times: source-sink is best!

- Buffers very well against internal fluctuations (steep slope) but poorly against external fluctuations

• Long averaging times: quadratic decay is best!

- Buffers well against external noise but poorly against internal

fluctuations (shallow slope)• Intermediate averaging times: exponential decay is best! Good compromise for both internal & external fluctuations

Saunders & Howard, Phys Rev E (2009)

Mechanism of gradient formation• Used parameters inspired by the Bicoid gradient

• Analysis assumes that gradients are generated by localised production with global diffusion/degradation

• New evidence that protein concentration gradients may arise from underlying mRNA gradient (Spirov et al, Development 2009)

• No consensus on underlying mechanism for Bicoid gradient formation

• Can we test our ideas without such a framework?

LLxBB //19.0exp1

)/(11 LfBB ind

nn

Lx

Lx

aB

19.019.0 0

)/( 0 Lxgaa indnn

Fitting the data without a model!• Assume a profile

where

• Identified a mutation in Bicoid cofactor dCBP (nej embryos)

• Profile altered and well fitted by

whereHe, Saunders, Wen, Cheung, Jiao, ten Wolde, Howard, Ma, submitted (2009)

Bicoid staining data

.22

int22

1

2

1

12

BB

BBy

BB

BB meas

ind

ind

,)('/

xBLB

LWBcd

,22

int

2

0

0

2

0

02

22

BB

BB

yy

yyny

aa

BB meas

indn

indn

Fitting the fluctuations• Fluctuations in staining intensity:

• Convert to positional error:

• Similarly for nej embroys

Is the Bicoid gradient precise?• Now compute error in Hb domain boundary in wt vs nej

• Precision off by factor of 2 (probably due to gap gene interactions)

• Perturbation in shape compromises precision of positional information

• What about optimising Λ=λ/L?

• Easy to calculate error as function of Λ

• Λopt≈0.12 compared to Λmeas≈0.18

• Bicoid gradient is highly precise!

Theory: WHb/L = 0.021±0.011 (wt) and WHb/L = 0.039±0.012 (nej)

Experiments: WHb/L = 0.011±0.003 (wt) and WHb/L = 0.022±0.005 (nej)

Interpreting gradients in pre-steady-state

• Can precision be improved by using pre-steady-state interpretation?

• Yes, according to Bergmann et al, PLoS Biology (2007)

• But this analysis only considered external fluctuations in J

• If internal and time-averaging window fluctuations included…

• … advantage evaporates (at least for Bicoid)

• Unlikely that Bicoid is interpreted in pre-steady-state

• Is possible for morphogens that are not direct transcription factors

Saunders and Howard Phys Biol (2009)

• Analysed effects of noise on simple gradient-forming mechanisms• Relevant to developmental (and cell) biology

• Positional information as an optimisation problem• Two ways to optimise:

– Optimise kinetic parameters (i.e. vary the decay length)– Optimise the overall shape of the profile (i.e. exponential vs

power law)

Conclusions

• Design principle: evolution optimises morphogen gradients to give maximally precise positional information

• Evidence from Bicoid that gradient is highly precise and optimised

• Interpreting morphogen gradients in pre-steady-state is problematic

AcknowledgementsFilipe Tostevin (Imperial, now Amsterdam)Timothy Saunders (John Innes Centre)Pieter Rein ten Wolde (AMOLF, Amsterdam)Feng He, Ying Wen, David Cheung, Renjie Jiao, Jun Ma (Cincinnati)

£££:

Postdoc position available! Please contact me at martin.howard@bbsrc.ac.uk

Calculating ind

22

0

1( ( , )) ( ( , ) ( , ) )n

i

x x i t x i tn

2 2 2( ( , )) ( ( ,0)) 1 (1 ) ( , )t

t tx x C x t dtt

22 2( ( ,0)) ( ,0) ( ,0)x x x

• Quantity of interest:

with τ=nδt

• Split into “diagonal” and “off-diagonal” components and take continuum limit in time:

2

22

( , ) ( ,0) ( ,0)( , )

( ,0) ( ,0)

x t x xC x t

x x

( ) exp( )C t t

/ 2

( )( ) exp( )(4 )

d

d

xC t tDt

22

2

1 ( ) 4ln2 ( )ind

xD x

where and

• Calculate correlation function for diffusion/degradation process

for Dt << (Δx)2

for Dt >> (Δx)2

Compute integral for μτ>>1

Müller-Krumbhaar and Binder, J. Stat. Phys. (1973)

~

• w=constant at short averaging times?

• At low densities, <n(xT)> « 1

• For short averaging times, everywhere there is a protein, n>n(xT)!

• Crossing distribution follows particle distribution

Short averaging times

1( ) exp( / )p x dx x dx

0

( )L

x xp x dx 2 2 2

0

( ) ( )L

w x x p x dx

• When is time-averaging beneficial?• When on average we have at least one protein at xT

• Average distance between particles at xT:

• Average time to diffuse this distance:

Crossover time

1/~ dl

2 exp( / )TxlD J

d=2

How does optimal gradient model with general n depend on time?

• Where n is the power in the morphogen decay term:

)(2 xJD nt

Results qualitatively unchanged