First use of bioRxiv: Reconstructing Promoter Activity From Lux Bioluminescent Reporters

Today I have made a new publication foray and submitted a manuscript to bioRxiv. This is the main paper to have come out of work on our BBSRC Lux grant. We are yet to find a peer-review home – but one of our co-authors has already had a conversation with someone who wants to use the method – so it was time to put the manuscript out there while we continue with the peer-review process. R code and Biomodels submission will follow. The manuscript details are:

Model Parameter estimation for Predictive Medicine

I am delighted to have been invited as a tutor to the Wellcome-Trust funded course on Model Parameter Estimation for Predictive Medicine organized by Sara Jabbari and Joanne Dunster. The course is being held from 4th-7th July at the University of Birmingham and is jointly run with the University of Nottingham.

The course promises to be extremely interesting, covering classical approaches to fitting dynamical models to data along with its main focus on Bayesian approaches, especially MCMC. Particularly pleased to be teaching alongside my colleagues Simon Preston and Theodore Kypraios.

Fitting models to data is a crucially important skill that has been somewhat neglected in mainstream mathematical biology and medicine so it is really great that Sara and Joanne have organized this and that Wellcome Trust has funded it. Do sign up!

New publication: A Bayesian approach to analyzing phenotype microarray data enables estimation of microbial growth parameters

Delighted to say that our first paper of 2016 is published. Matthias’s Biolog paper is now online-ready with the Journal of Bioinformatics and Computational Biology. This is also the first output from our Biolog grant – with a second paper detailing our newer software and analysis being planned.

Gerstgrasser M, Nicholls S, Stout M, Smart K, Powell C, Kypraios T and Stekel DJ. 2016. A Bayesian approach to analyzing phenotype microarray data enables estimation of microbial growth parameters. J Bioinform Comput Biol. DOI: 10.1142/S0219720016500074

Abstract

Biolog phenotype microarrays (PMs) enable simultaneous, high throughput analysis of cell cultures in different environments. The output is high-density time-course data showing redox curves (approximating growth) for each experimental condition. The software provided with the Omnilog incubator/reader summarizes each time-course as a single datum, so most of the information is not used. However, the time courses can be extremely varied and often contain detailed qualitative (shape of curve) and quantitative (values of parameters) information. We present a novel, Bayesian approach to estimating parameters from Phenotype Microarray data, fitting growth models using Markov Chain Monte Carlo (MCMC) methods to enable high throughput estimation of important information, including length of lag phase, maximal “growth” rate and maximum output. We find that the Baranyi model for microbial growth is useful for fitting Biolog data. Moreover, we introduce a new growth model that allows for diauxic growth with a lag phase, which is particularly useful where Phenotype Microarrays have been applied to cells grown in complex mixtures of substrates, for example in industrial or biotechnological applications, such as worts in brewing. Our approach provides more useful information from Biolog data than existing, competing methods, and allows for valuable comparisons between data series and across different models.

New Publication: Analysis of Occludin Trafficking, Demonstrating Continuous Endocytosis, Degradation, Recycling and Biosynthetic Secretory Trafficking

We are delighted that our work with Josh Rappoport‘s laboratory, supported by the Birmingham-Nottingham Strategic Collaboration Fund, has led to successful publication. Well done to all, especially Sarah Fletcher who carried out all the experimental work, Mudassar Iqbal who did the model data fitting, and Sara Jabbari who has helped out both Mudassar with the modelling and Sarah with sorting out journal requirements.

Fletcher, S.J., Iqbal, M., Jabbari, S., Stekel, D.J. and Rappoport, J.Z. 2014. Analysis of Occludin Trafficking, Demonstrating Continuous Endocytosis, Degradation, Recycling and Biosynthetic Secretory Trafficking. PLoS ONE DOI: 10.1371/journal.pone.0111176.

Abstract

Tight junctions (TJs) link adjacent cells and are critical for maintenance of apical-basolateral polarity in epithelial monolayers. The TJ protein occludin functions in disparate processes, including wound healing and Hepatitis C Virus infection. Little is known about steady-state occludin trafficking into and out of the plasma membrane. Therefore, we determined the mechanisms responsible for occludin turnover in confluent Madin-Darby canine kidney (MDCK) epithelial monolayers. Using various biotin-based trafficking assays we observed continuous and rapid endocytosis of plasma membrane localised occludin (the majority internalised within 30 minutes). By 120 minutes a significant reduction in internalised occludin was observed. Inhibition of lysosomal function attenuated the reduction in occludin signal post-endocytosis and promoted co-localisation with the late endocytic system. Using a similar method we demonstrated that ~20% of internalised occludin was transported back to the cell surface. Consistent with these findings, significant co-localisation between internalised occludin and recycling endosomal compartments was observed. We then quantified the extent to which occludin synthesis and transport to the plasma membrane contributes to plasma membrane occludin homeostasis, identifying inhibition of protein synthesis led to decreased plasma membrane localised occludin. Significant co-localisation between occludin and the biosynthetic secretory pathway was demonstrated. Thus, under steady-state conditions occludin undergoes turnover via a continuous cycle of endocytosis, recycling and degradation, with degradation compensated for by biosynthetic exocytic trafficking. We developed a mathematical model to describe the endocytosis, recycling and degradation of occludin, utilising experimental data to provide quantitative estimates for the rates of these processes.

Inferring the error variance in Metropolis-Hastings MCMC

One of the great joys of working with two talented post-docs in the research group – Mike Stout and Mudassar Iqbal – as well as a great collaboration with Theodore Kypraios, is that they are often one step ahead of me and I am playing catch-up. Recently, Theo has discussed with them how to estimate the error variance associated with the data used in Metropolis-Hastings MCMC simulations.

The starting point, usually, is that we have some data, let us say y_i for i=1, \cdots, n, and a model – usually, in our case, a dynamical system – which we are trying to fit to the data. For any given set of parameters \theta, our model will provide estimates for the data points that we will call \hat y_i. Now, assuming uniform Gaussian errors, our likelihood function L(\theta) looks like:

L(\theta) = \prod_{i=1}^n \frac{1}{\sqrt{2 \pi\sigma^2}}e^{-\frac{1}{2}(\frac{y_i - \hat y_i}{\sigma})^2}

where \sigma^2 is the error variance associated with the data. Now, when I first started using MCMC, I naively thought that we could use values for \sigma^2 provided by our experimental collaborators, and so we could use different values of \sigma^2 according to how confident our collaborators were in the measurements, equipment etc. What I found in practice was that these values rarely worked (in terms of convergence of the Markov chain) and we have had to make up error variances using trial and error.

So I was delighted when I heard that Theo had briefed both Mike and Mudassar about a method for estimating the error variance as part of the MCMC. Since I have not tried it before, I thought I would give it a go. I am posting the theory and some of my simulations, which are helpful results.

Theory

The theory behind estimating \sigma^2 is as follows. First, set

\tau = \frac{1}{\sigma^2}

We can then re-write the likelihood, now for the model parameters \theta and also the unknown value \tau, as

L(\bf{\theta}, \tau) = \frac{\tau^{(n/2)}}{\sqrt{2 \pi}^n}e^{-\frac{\tau}{2}\sum_{i=1}^n(y_i - \hat y_i)^2}

Now observe that this has the functional form of a Gamma distribution for \tau, as the p.d.f. for a Gamma distribution is given by:

f(x; \alpha, \beta) = \frac{\beta^\alpha}{\Gamma(\alpha)}x^{\alpha-1}e^{-\beta x}

So if we set a prior distribution for \tau as a Gamma distribution with parameters \alpha and \beta, then the conditional posterior distribution for \tau is given by:

p(\tau | \theta) \propto \tau^{(n/2)+ \alpha - 1}e^{-\tau(\frac{1}{2}\sum_{i=1}^n(y_i - \hat y_i)^2+\beta)}

We observe that this is itself a Gamma distribution, with parameters \alpha \prime = \alpha + n/2 and \beta \prime = \beta + \frac{1}{2} \sum_{i=1}^n (y_i - \hat y_i)^2. Thus the parameter \tau can be sampled with a Gibbs step as part of the MCMC simulation (usually using Metropolis-Hastings steps for the other parameters).

Simulations

The simulations I have run are with a toy model that I use a great deal for teaching. Consider a constitutively-expressed protein that is produced at constant rate k and degrades (or dilutes) at constant rate \gamma per protein. A differential equation for protein concentration P is given by:

\frac{dP}{dt} = k - \gamma P

This ODE has the closed form solution:

P = \frac{k}{\gamma} + (P_0 - \frac{k}{\gamma}) e^{-\gamma t}

where P_0 is the concentration of protein at t=0. For the purposes of MCMC estimation, mixing is improved by setting P_1 = \frac{k}{\gamma} so that the closed form solution is:

P = P_1 + (P_0 - P_1) e^{-\gamma t}

Some data I have used for teaching purposes comes from the paper Kim, J.M. et al. 2006. Thermal injury induces heat shock protein in the optic nerve head in vivo. Investigative ophthalmology and visual science 47: 4888-94. The data is quantitative Western blots of Hsp70 in the optic nerve of rats, as induced by laser damage. (Apologies for the unpleasantness of the experiment):

Time / hours Protein / au
3 1100
6 1400
12 1700
18 2100
24 2150

The aim is to use a Metropolis-Hastings MCMC, together with a Gibbs step for the \tau parameter, to fit the data. The issue that immediately arises is how to set the parameters \alpha and \beta. This may seem arbitrary, but it is already better than choosing a value for \sigma^2, as the Gamma distribution will exploring of that parameter. For my first simulation, I thought that \sigma = 100 would be sensible (this turned out to be a remarkably good choice, as we will see). So I set \alpha = 0.01 and \beta = 100 and lo and behold, the whole MCMC worked beautifully. (Incidentally, I used independent Gaussian proposals for the other three parameters, with standard deviations of 100 for the P_0 and P_1 and standard deviation of 0.01 for \gamma. These parameters were forced to be positive – Darren Wilkinson has an excellent post on doing that correctly. Use of log-normal proposals in this case leads to very poor mixing, with the chain taking some large excursions for the P_1 and \gamma parameters).

histograms

The median parameter values are P_0 = 786, P_1 = 2526, \gamma = 0.0686 and \tau = 0.000122. The latter corresponds to \sigma = 90.6. With these values, we can see a good fit to the data: below are plotted the data points (in red), the best fit (with median parameter values) in blue, and model predictions from a random sample of 50 parameter sets from the posterior distribution in black.

fit

Considerations

However, some questions obviously arise: how sensitive is this procedure to choices of \alpha and \beta? I will confess: I use Bayesian approaches fairly reluctantly, being more comfortable with classical frequentist statistics. What I like about Bayesian approaches are firstly the description of unknown parameters with a probability distribution, and secondly the availability of highly effective computer algorithms (i.e. MCMC). What makes me uncomfortable is the potential for introducing bias through the prior distributions. So I have carried out some investigations with different values of \alpha and \beta. In particular, I wanted to know: (i) what happens if I keep the mean (equal to \alpha / \beta) the same but vary the parameters? (ii) what happens if I vary the mean of the distribution? The table below summarizes positive results:

alpha beta P0 P1 gamma sigma
0.01 100 786 2526 0.0686 90.6
1 10000 747 2428 0.0795 98.0
0.0001 1 797 2533 0.0681 96.3
0.1 10 822 2603 0.0623 97.9
0.001 1000 760 2455 0.0758 94.8
1 1 792 2539 0.0676 64.9
0 0 805 2565 0.0653 98.3

As you can see (please ignore the last line for now), the results are robust to a very wide range of \alpha and \beta, even producing a good estimate for \sigma when that estimate is a long way from the mean of the prior distribution. But then we can make the following observation. Consider the sum of squares for a ‘best-fit’ model, for example using the parameters for the first row (this is 12748). So as long as \alpha \ll n/2 and \beta \ll 12748/2, the prior will introduce very little bias. But if you try to use values of \alpha and especially \beta very much larger than an estimated sum of squares from well-fitted model parameters, then things might go wrong. For example, when I set \alpha = 1 and \beta = 10^6 then my MCMC did not converge properly.

This leads to my final point, and the final row in the table. Would it be possible to remove prior bias altogether? If you look at the marginal posterior for \tau, we observe that if we set \alpha = \beta = 0, we obtain a Gamma distribution, whose mean is precisely the error variance, as, in this case,

\frac{\beta \prime}{\alpha \prime} = \frac{\sum_{i=1}^n(y_i - \hat y_i)^2}{n}

The algorithm should work perfectly well sampling from this Gamma distribution, and indeed it does, producing comparable results to when an informative prior is used.

Conclusions

In summary, I am happy to conclude that this method is good for estimating error variance. Clear advantages are:

  1. It is simple to implement and fairly fast to run – adding a Gibbs step is no big deal.
  2. It is clearly preferable to making up a fixed number for the error variance – which was what we were doing before.
  3. The prior parameters allow you to make use of information you might have from experimental collaborators on likely errors in the data.
  4. The level of bias from the priors is relatively low, and can be eliminated altogether.

Job opportunity: two-month postdoctoral position in mathematical modelling / inference

Research Associate/ Fellow

Reference
SCI10709
Closing Date
Friday, 8th February 2013
Job Type
Research & Teaching
Department
School of Biosciences – Division of Agricultural & Environmental Science, Multidisciplinary Centre for Integrative Biology
Salary
£24766 to £29541 per annum depending on skills and experience, minimum £27,854 per annum with relevant PhD.

This full-time post is available on a fixed term contract for a period of two months.

Applications are invited to join a highly motivated multi-disciplinary team of research scientists working the Universities of Nottingham and Birmingham. The successful candidate will join a jointly funded project to carry out modeling of occludin trafficking during epithelial polarization and wound healing. The post could be located either in the School of Biosciences at the University of Nottingham’s Sutton Bonington Campus, or at the School of Biosciences at the University of Birmingham.

The work will include (i) developing a mathematical models (using ODEs) to describe the turnover of occludin protein in the cell as well as the kinetic trafficking of occludin between cellular compartments; (ii) to estimate model parameter values from experimentally derived data using Monte Carlo Markov Chain approaches; and (iii) to iteratively improve the model, with cycles of model and data comparison, in order to provide greater certainty about the important mechanisms involved that can explain the experimental data. Other duties will include contributing to publication of this research in peer-reviewed journals, contributing to writing of research grant applications, and generally collaborating between disciplines and institutions.

The successful candidate must have a PhD or equivalent in mathematical modelling or statistics or a related area. Research experience within a mathematical biology or systems biology research area would be desirable but not essential. Candidates must to be able to demonstrate excellent mathematical ability, especially in the areas of ordinary differential equations and statistical analysis of data; experience of application of these skills to biological research would be desirable. Candidates must also be able to evidence excellent computing skills in a suitable environment (e.g. R or Matlab). Excellent English language oral and written communication skills are also essential. This post will require the person appointed to be able to work independently and as part of a multi-disciplinary team, to be motivated, flexible and willing to learn.

Full details, including how to apply, can be found on the University of Nottingham’s vacancy system.

Informal enquiries may be addressed to Dr Dov Stekel, email: dov.stekel@nottingham.ac.uk or Dr Josh Rappoport, email: j.rappoport@bham.ac.uk.

Birmingham-Nottingham Strategic Collaboration Fund awarded

We are delighted that the Universities of Birmingham and Nottingham have awarded a Strategic Collaboration Fund award to Josh Rappoportand myself. The project is titled “Experimental analysis and modeling of occludin trafficking during epithelial polarization and wound healing”. We will be looking at the modelling end, developing an ODE model and fitting to data from Josh’s lab using Monte Carlo techniques. The funding awarded is £20,000 and we will be looking for a short-term post-doctoral research fellow for a two month period to carry out the work – to start at some point in 2013 (further details to be posted).

We are very much looking forward to collaborating with Josh and others on this project.