Martin Howard

1. Modelling Noisy Concentration Gradients in Developmental Biology Martin Howard Dept of Systems Biology John Innes Centre, UK

2. 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

3. 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 • Very simple model, but potentially still biologically relevant! • Morphogens Bicoid, Dpp, Wingless in Drosophila all have exponential profiles

4. Example: Bicoid! • Intensively studied example: Bicoid protein in fruit fly Drosophila melanogaster • Gradient well fitted by simple exponential • Bicoid drives very precise gene expression of hunchback! Houchmandzadeh et al. Nature (2001)

5. Noisy gradients Noise can be due to external or internal variation External fluctuations give cell-to-cell or embryo-to-embryo variability Internal fluctuations affect accuracy within a single cell/embryo • Internal fluctuations give limit to precision of positional information: can we calculate this?!

6. 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: • Now convert fluctuations in density to error in positional information, given by a width: Tostevin, ten Wolde, Howard PLoS Comput Biol (2007) See also Gregor et al, Cell (2007)

7. 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

8. Reducing the imprecision: time-averaging • Integrating for time twe can make Nt=t/tind independent measurements • tind~(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! Spatial analog of Berg-Purcell result (1977)

9. Simulations For xT=2μm, w is minimised at λ=2μm Data collapse for long averaging times:

10. Role of detector size In d=2, w is independent of Δx (up to log correction) • ReducingΔx 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!

11. Noisy gradients Noise can be due to external or internal variation External fluctuations give cell-to-cell or embryo-to-embryo variability Internal fluctuations affect accuracy within a single cell/embryo • So now we understand internal fluctuations, but what about the effect of external fluctuations?

12. Combining External and Internal Fluctuations Saunders & Howard, Phys Rev E (2009) • Calculate imprecision W in positional information due to embryo-to-embryo fluctuations • Focus on fluctuations δJin 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?

13. Maximising precision 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!

14. 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

15. Quadratic degradation 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? Barkai et al. Dev. Cell (2003)

16. 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 asd=3 is above the upper critical dimension of dc=2

17. Maximising precision in the quadratic decay model 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

18. 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?!

19. 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) Saunders & Howard, Phys Rev E (2009) • Intermediate averaging times: exponential decay is best! Good compromise for both internal & external fluctuations

20. 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?

21. 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 • where Bicoid staining data He, Saunders, Wen, Cheung, Jiao, ten Wolde, Howard, Ma, submitted (2009)

22. Fitting the fluctuations • Fluctuations in staining intensity: • Convert to positional error: • Similarly for nej embroys

23. 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)

24. 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)

25. Conclusions • 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) • 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

26. Acknowledgements Filipe 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

27. Calculating tind Müller-Krumbhaar and Binder, J. Stat. Phys. (1973) • Quantity of interest: withτ=nδt • Split into “diagonal” and “off-diagonal” components and take continuum limit in time: where and • Calculate correlation function for diffusion/degradation process for Dt << (Δx)2 Compute integral for μτ>>1 ~ for Dt >> (Δx)2

28. Short averaging times • 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

29. Crossover time • 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: d=2

30. How does optimal gradient model with general n depend on time? • Where n is the power in the morphogen decay term: Results qualitatively unchanged