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Modelling Noisy Concentration Gradients in Developmental Biology. Martin Howard. Dept of Systems Biology John Innes Centre, UK. Position determination in biology. How to measure position in biological systems? - PowerPoint PPT Presentation
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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 x
xD
• 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
xDw
J x
positional information, given by a width:
Tostevin, ten Wolde, Howard PLoS Comput Biol (2007)
See also Gregor et al, Cell (2007)
)(
)0,(0
T
T
x
xw
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)( , )
xx
N
2exp
( ) 2T
d
xw k
J 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!
2exp
( ) 2T
d
xw k
J 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?
J
JW
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 )/(
J
Je
xNJ
DDkTx
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 )(
)(xx
Ax
D
A6
3
12
0
12
JD
x
J
JxWquad
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
)/(
J
Jx
x
xx
xNJ
DDk 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
JxLxL
xNJ
DDkTT
spatss
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?
LL
xBB //19.0exp1
)/(11 LfBB ind
nn
L
x
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
int
22
1
2
1
1
2
B
B
B
By
B
B
B
B measind
ind
,)('
/
xB
LB
L
WBcd
,22
int
2
0
0
2
0
02
22
B
B
B
B
y
y
yy
ny
a
a
B
B measindn
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 [email protected]
Calculating ind
2
2
0
1( ( , )) ( ( , ) ( , ) )
n
i
x x i t x i tn
2 2 2( ( , )) ( ( ,0)) 1 (1 ) ( , )
t
t tx x C x t dt
t
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 t
Dt
22
2
1 ( ) 4ln
2 ( )ind
x
D 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( / )Txl
D 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