1. Matching Models to Data in Modelling Morphogen Diffusion
Wei Liu and Mahesan Niranjan
School of Electronics and Computer Science
University of Southampton
United Kingdom
{wl08r,
Hi everyone, I am wei liu. I come from uni of southampton
Glad to be here to introduce my work to you.
I am going to talk about matching models to data in modelling morphogen diffusion.

2. Drosophila Small flies Length of embryo: 500 Short generation time
Genetics
Common model
Before that, let me introduce the background of my work.
Drosophila is small fly with yellow or brown body and red eye. The embryo of drosophila is about 500 micrometers . These flies have short generation time.
It has been heavily used in research of genetics and it is a common model in development biology.
flydatabase

3. Drosophila Development
After Fertilization
Nuclear division
No cytoplasm cleavage
Syncytium
Pattern formation
Cellular blastoderm
Body axes segment boundaries
Lets look at dro development.
In the early drosophila embryo development,
Following fertilization, the nuclear division begins and There is no cleavage of cytoplasm.
And we call this common cytoplasm with multi nuclei as syncytium.
It also allows morphogen gradient to play a key role in pattern formation.
After about 3 hours , cell walls develop and is called cellular blastoderm.
then The major body axes and segment boundaries are determined.
Finally, we get the adult fly.
from LIFE: The Science of Biology, Purves et al, 1998

4. Outline Passive diffusion models for spatial patterns establishment
Constant supply
Bicoid morphogens
Constant supply followed by exponential decay
Models vs Measured data
Parameter estimation
a brief overview of my talk.
The passive diffusion models
of morphogen proteins translated from maternally deposited messenger
RNAs are usually used for spatial patterns establishment.
Such model always assume a constant supply of morphogens
By working in bicoid morphogen in Drosophila, we note that this constant assumption is not realistic because mRNA is known to decay after a certain time.
We solve the models with combined supply numerically and compare the output with the measured data from flydatabase and estimate the parameters used in the model. These parameters can be assigned sensible values

5. What is the morphogen? Molecules in multi-cellular organism.
Establishing spatial patterns of gene expression.
Cells far from source: low level
Cells close to source: high level
Subdivided into different types.
Different cells organise to form different organs
Firstly, I want to introduce Morphogen
Alan turing in 1952 discrebed a reaction diffusion system of morphogen. He discussed a possible mechanism that the genes may determine the structure of organism.
Morphogens are a class of molecules in multi-cellular organism.
They can provide concentration gradient from a localized source to establish spatial patterns of gene expression in embryo development.
The cells far from the source will receive low level of morphogen concentration while cells close to the source will receive high level of concentration.
Therefore, the cells can be subdivided into different types by their position to the source.
These various cells form different organs during development.
Turing, A.: The chemical basis of morphogenesis. Philosophical Transactions of
the Royal Society B 237(641) (1952) 37–72

6. Bicoid Morphogen Drosophila body plan and position information.
Contributing to set up the anterior posterior axis.
Controlling cells fate along 70% of this axis.
There are some maternal genes which have contributions to drosophila body plan and position information.
In early drosophila embryo development, bicoid is a kind of morphogen
that has contribution to set up the anterior-posterior axis [3] and it controls
cells fate along 70% of this axis [4].

7. Bicoid Morphogen Concentration
Here I want to show how the bicoid protein concentration gradient establish .
1. Initially, The maternally provided bicoid mRNA is deposited in the anterior pole of the oocyte.
2. Then mRNA translate to Bicoid proteins after fertilization.
3. After some time, Bicoid proteins diffuse from localized source. And then these proteins form a concentration gradient from anterior to posterior of the embryo. This gradient will induce gap genes expression.
4. After a certain time, the mRNA will decay.
Anterior part
Posterior part

8. Reaction-diffusion Equation
The reaction-diffusion equation of single-morphogen concentration system is below:
M(x,t) is morphogen concentration
S(x,t) is a general source term at the anterior pole
D is diffusion constant
is half-life of the morphogen protein
This reaction diffusion equation. This is a spatio-temporal one dimension diffusion model.
where M(x, t) is the morphogen concentration as a function of space and time.
D is the diffusion constant
Tau_p is the half-life of the morphogen protein
S(x, t) is the source at the anterior end.

9. Constant Source Usual assumption
The usual assumption in solving this model is
that the source is constant:
S0 here is the production rate,
(x) is the Kronecker delta function and (t) is
Heaviside step function.
Therefore, we have a point source at x = 0,
Then there is constant supply when t increasing.

10. New Source Model
Here, we work with a source model which has a constant part during which
the maternal mRNA is kept stable, followed by an exponentially decaying part,
due to maternal mRNA decaying from about cleavage cycle 12 of the developing
embryo.
There is one paper in 1998 found that mRNA will decay after cycle 12.
Surdej, P., Jacobs-Lorena, M.: Developmental regulation of bicoid mrna stability is mediated
by the first 43 nucleotides of the 3’ untranslated region. Molecular and Cellular Biology
18(5) (1998) 2892–2900

11. Widely used Model with constant source
This figure shows the solution to the reaction diffusion model of morphogen concentration
With the constant source
This intensity profile is jointly in time and along the length of embryo. The right
panel shows the constant supply of bicoid proteins at the anterior end of the embryo.
This is the widely used model. which at steady state sets up an exponential profile.
We got this solution with numerical method using pdepe Toolbox

12. The more realistic model
shows the solution to the model, which we thinks as more realistic,
in which the
source is a combination of a constant supply followed by an exponential decay because the maternally
deposited bicoid mRNA is decaying.
After peak value, we call this part post peak
The intensity of this solution is different from former constant one. The concentration is increasing and then decreasing in the same position of embryo.

13. Measured Data Ⅰ Flyex Database to estimate parameters of the model.
Measured data: one dimension Bicoid integrated data in nuclear cleavage cycle 14A.
Cycle 14A : 50 mins in duration; 8 equal temporal classes; 6.5 mins each class.
We used FlyEx database [8] to estimate parameters of the model.
In Flyex Database [8], bicoid integrated data in nuclear cleavage cycle 14A in
one- dimension is used as measured data. Cycle 14A is nearly 50 mins
in duration and is divided into 8 equal temporal classes of 6 mins duration.
Andrei Pisarev, Ekaterina Poustelnikova, Maria Samsonova, John Reinitz (2009) FlyEx, the quantitative atlas on segmentation gene expression at cellular resolution. Nucl. Acids Res.; 37: D560 - D566. Ekaterina Poustelnikova, Andrei Pisarev, Maxim Blagov, Maria Samsonova, and John Reinitz (2004). A database for management of gene expression data in situ

14. Measured Data Ⅱ 1D integrated data – cycle11- cycle 14A (1-8 classes)
This figure shows intensities of bicoid morphogens in cycle 14a for different classes. These are one dimensional data
From FlyEx Database:

15. Measured Data Ⅲ
Integrated 2D patterns – reconstructed image 14A-2

16. Comparison of model based and measured data of bicoid intensities
130
136
142
148
154
160
shows intensities of morphogen diffusion model output with a constant
source followed by an exponentially decaying source, and the measured data
from FlyEx.
During this period, the bicoid concentration begins to decay due to the decay of the
source mRNA and the diffused protein.
It shows that there is a good match between these two matrix.
Here is the 8 temporal classes.
in the post-peak stages of morphogen profile, jointly in space and time. To the
best of our knowledge, these stages have not attracted interest in the literature,
and the popular model with a constant morphogen source is clearly incorrect in
these stages.
166
172
178
Time (mins)

17. Matching Parameter Values to Data Ⅰ
Squared error between model output and measured intensities to evaluate error.
Parameters estimation:
In this work, we used the squared error between model output and measured intensities to evaluate error in estimating the parameters.
where T1 and T2 were the boundaries of cleavage cycle 14A for which data,
Md(x, t), was available at eight uniformly sampled time points
We need to estimate 4 parameters:
The errors are computed during cycle 14A, holding three of the four parameters at their best estimates
from literature and varying the fourth.
Molecular and cellular biology 1998
Diffusion constant D = ,
The time mRNA starts to decay = 120mins,
mRNA half-life = 29mins
Bicoid protein half-life = 111mins.

18. Matching Parameter Values to Data Ⅱ
The errors in the joint space of diffusion constant and maternal mRNA
decay onset time.
The errors in the joint space of diffusion constant and maternal mRNA
decay onset time.
and achieves a minimum in the range of sensible values.

19. Finding Optimal Values for all the Parameters Ⅲ
Best combination of parameter values simultaneously.
Diffusion constant D = ,
The time mRNA starts to decay = 118mins,
mRNA half-life = 28.4mins
Bicoid protein half-life = 120mins.
We also searched for the best combination of parameter values at the same time.
This search resulted in values closed to the results
obtained previously.
Bergmann et al. [4] suggest a value for diffusion constant in the range
μm2/s.
When degradation starts later than
140mins or earlier than 110mins, the error increases significantly.
Bergmann et al. [4] suggested values for p should
be higher than the range mins
Our estimate of bicoid mRNA half-life is 29mins, which is nearly 1/4 of bicoid
proteins decaying time from our model.

20. Conclusion and Future Work
Widely used model with a constant source is unrealistic.
By matching models output to data.
The single measurements without uncertainties.
Developing a stochastic model for a population of embryos i.e. Master equation model
Developing data driven model for embryo spatio- temporal data i.e. Kriged Kalman Filter
Widely used models of a constant supply of morphogens at the source is not realistic for a number of reasons.
i.e. decay of the source mRNA
after a certain time, using the morphogen bicoid as an example.
how parameters of the diffusion model can be calculated.
In the present study we have used single measurements of profiles from the FlyEx
database, which do not have uncertainties of the measurements quantified. The
next step in this work would be to acquire uncertainties in bicoid profile measurements,
arising from a distribution across a population of embryos, formulate the
estimation problem in a probabilistic setting, and carry out posterior inference
along the lines in [11].

21. Thanks !

22. Kriged Kalman Filter KF – linear Gaussian state space model
KKF –modelling spatio-temporal data (Mardia et. al 1998)
KF is used for the linear gaussian state space model. Under the assumption that all the parameters are known. However, this state space model is only for temporal dynamic.
KKF proposed by Mardia can be used to model spatio-temporal data. And we want to use kkf for our embryo data which also spatio-temporal data. Base on this model, we can inference the underlying state or learning parameters. Of course this is still a challenge for us to apply this model. There are many difficults. Ie, the time point is limited and it will cause over-fitting problem.

23. FlyEx Database Step 1 : Data Acquisition
Acquisition of quantitative data on gene expression in individual embryos
Step 2 : Data Registration
Excision of 10 % stripe of quantitative data
Feature extraction
Registration
Step 3: Data Normalization
Rescaling of data to bring data to unified standard form with a zero background
Step 4: Data Averaging
Construction of the integrated pattern of each gene expression
Flyex database stores quantitative data on gene expression in segmentation genetic network in Drosophila melanogaster
Andrei Pisarev, Ekaterina Poustelnikova, Maria Samsonova, John Reinitz (2009) FlyEx, the quantitative atlas on segmentation gene expression at cellular resolution. Nucl. Acids Res.; 37: D560 - D566. Ekaterina Poustelnikova, Andrei Pisarev, Maxim Blagov, Maria Samsonova, and John Reinitz (2004). A database for management of gene expression data in situ . Bioinformatics, 20: