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**A review of:** **Julian Lewis (2003) //Autoinhibition with Transcriptional Delay: A Simple Mechanism for the Zebrafish Somitogenesis Oscillator,// Current Biology Vol. 13, 1398-1408** //by: Remi Verhoeven//

The original publication can be found here DOI 10.1016/S0960-9822(03)00534-7

This review is created for educational purposes for the course: //'Systems Biology'// by Natal van Riel at the Eindhoven University of Technology, The Netherlands.

toc include component="pageList" hideInternal="true" homeAtTop="on" limit="10" =Synopsis= This wiki-page is a review on the aforementioned paper by Julian Lewis in 2003. This review coveres the paper in detail and reproduces its results. After the introduction to the topic, the methods used to model somitogenesis and the results of these models, a discussion section will critique the paper; it's usability, methods and results. A visual comparison between the results in the paper and the results obtained from the analysis can be viewed here.

Futhermore, the matlab-files used in the analysis of the paper will be presented as a seperate page, but are also presented in-line in this page.

This webpage is viewed best on a screen with a resolution of 1280x800 or higher.

=Introduction= Based on its genetic accesibility, the Zebrafish has become the Drosophila of the vertebrae developmental biology. Besides genetic and molecular research in to the zebrafish, extremely valuable research is ongoing in other fish models to test the generality of findings in zebrafish biology. One popular topic in zebrafish biology is somitogenesis; the embryonic development of trunk segmentation. The first somites in zebrafish embryo appear approximately 10.5 hours after fertilization. Additional somites are produced in 30 minute intervals in a bilaterally symmetric, anterior to posterior wave until a total of about 30 somite pairs bracket the notochord. Several models are created that pose the existence of a molecular clock that functions with temopral periodicity. This clock would translate a smooth maturational or positional gradient into a spatially periodic pattern, allowing somitogenesis to occur at regular intervals in successive, uniformly sized blocks of cells. The paper by Lewis describes a model that, by descibing an autorepression model for two linked oscillating genes (//her1// and //her7//), provides a mechanism for the intracellular oscillator.

In Lewis' paper, several statements are made on the oscillating behavior of somitogensis:
 * //deltaC//, //her1// and //her7// oscillate and code for parth of the Notch signaling pathway, or are regulated by Notch signaling (p.1398)
 * The mRNA levels of the these genes go up and down synchronously with a period of 30 min (p. 1398)
 * Salt-and-peper noise from individual //deltaC// expression shows evidence of necessary cell-cell communication via the Notch pathway (p. 1399)
 * //her1// and //her7// appear to regulate their own expression (p. 1399)

From these types of behavior, the following model assumptions arrise:
 * 1) There should be a negative feedback loop that regulates the oscillator (p. 1399)
 * 2) This feedback loop should be intracellular (p. 1399)
 * 3) The cells should be able to influence one another via the Notch pathway (p. 1399)
 * 4) There are delays between protein concentration and transcription, and translation and protein emergence (p. 1400,1401)

=Methods= To view all matlab code used to reproduce the results in the paper, click here.

Direct Autoinhibition of Gene Expression
Based on model assumptions 1,2 and 4, we can construct the following model for a single cell (figure 1):

__**equation (1)**__ dp(t)/dt = a·m(t - Tp) - b·p(t) dm(t)/dt = f(p(t - Tm)) - c·m(t)

with:

f(p) = k / (1 + p 2 /p 0 2 )

where, //p// is the number of protein molecules currently available, //m// is the number of mRNA molecules currently available, //a// is the rate of production of new protein molecules per mRNA molecule, //b// and //c// are decay reates of the protein and mRNA molecules respectively and //f(p)// is the rate of production of new mRNA molecules, which is assumed to be a decreasing function of the amount of protein //p//. The behavior is analysed with the following plausible values:
 * ~ parameter ||~ //her1/her7//-value ||~  ||~ unit ||
 * = a ||= 4.5 || || [protein molecules per mRNA molecule per minute] ||
 * = b ||= 0.23 || || [protein molecules per minute] ||
 * = c ||= 0.23 || [5][6] || [mRNA molecules per minute] ||
 * = k ||= 33 || [4] || [mRNA molecules per diploid cell per minute] ||
 * = p0 ||= 40 ||  || [molecules] ||

To estimate the delays, several assumptions/estimations are made:
 * RNA polymerase II moves along DNA at a rate of 20 nucleotides per second (p. 1402) [4]
 * An intron takes between 0.4 and 7.5 minutes to splice out (p. 1402)
 * There is an aditional delay of 4 minutes before the spliced mRNA comes in to the cytosol (p. 1402) [7]
 * Robisomes translate 6 nucleotides per second (p. 1402) [4]

These assumptions lead to the following ranges/values of the delay terms in equation (1): If the assumption is made that splicing out introns takes approximately 1 min, T mher1 = 12 min and T mher1 = 7.1 min. The matlab code for this (continuous) model is found here: The results to this analysis for T mher1 = 12 min and T mher1 = 7.1 min with and without attenuated protein synthesis can be found here.
 * ~ delay ||~ value ||~ unit ||
 * = T mher1 || [10.2, 31.5] || [min] ||
 * = T pher1 || 2.8 || [min] ||
 * = T mher7 || [5.9,20.1] || [min] ||
 * = T pher7 || 1.7 || [min] ||

Noisy Control of Gene Expression
If the assumption is made that the binding and dissociation of a gene regulatory protein to and from its site on DNA are not deterministic but stochastic, equation (1) will become a system of discrete differential equations with a similar time step to the stochastic process modeled by Lewis. There are several assumptions underlying this discrete interpretation of the differential equations:
 * the concentration (q) of the //her1// dimers is low compared to the concetration of the //her1// monomers; q is proportional to //p 2 //(supl. data)
 * transcripts are initiated at zero-rate where the repressor is bound and at maximal rate when it is not bound (supl. data)
 * time step (dt) is a finite step, not a infinitesimal (supl. data)

First-off, the binding and dissociation between the DNA is approach from a haploid 'point of view', this means that only one strain of DNA is taken in to account. A time series //X// is generated based on succesive probable states of binding and dissociation. These are obtained by means of the following equations:

P(n|m) = P(X(t+dt) | X(t)=m)
 * __equation (2)__**

with P(n|m) denoting the change to obtain //n// given //m//, //n// and //m// are either 0 or 1, P(X(t) | X(t-1)=a) = D(t) | D(t-1)=a, where

D(t) = k off / (k on + k off )

in which k off corresponds to the mean lifetime of the repressor bound state (p. 1403), and k on is a function of the concentration of the protein //q//. If the ploidity is assumed diploid instead, with two gene copies subject to regulation, this boils down to:

P(n|m) = P(XX(t+dt) | XX(t)=m)
 * __equation (3)__**

where //n// and //m// are eiterh 0,1 or 2. Using the same values as in the direct autoinhibition of gene expression for //a//, //b//, //c//, //k// and //p 0 //, the matlab code becomes:

The results of the analysis for T mher1 = 12 min and T mher1 = 7.1 min with and without attenuated protein synthesis can be found here.

Delta-Notch Communication in Adjacent Cells
To achieve synchronized oscillation between cells (obey assumptions 1,2,3 and 4), a connection between two cells should be made. The Notch signalling mechanism is an excellent candidate for this purpose. In this example, Lewis disregards stochastic effects and considers two adjacent cells with slightly different free-running oscillation periods, so that without cell-cell communication their cycles would desynchronize. To model the effect of Notch-mediated communication, there are some aditional assumptions that should be made:
 * //deltaC// expression is regulated in parallel with expression of //her1// and //her7// by Her1/Her7 protein acting directly on the //deltaC// promotor (p. 1404)
 * the lifetimes of mRNA and protein of //deltaC// are short (p. 1404)

The total delay T N in the Notch pathway is calculated by:

T N = T md + T pd + T exp + T NA
 * __equation (4)__**

where T md is the delay in initiation of transcription of //deltaC// to a mature mRNA molecule, T pd the delay from initiation to completion of synethesis of a DeltaC protein molecule, T exp the delay for delivery of the protein to the cells surface, and T NA the delay from the time when functional DeltaC protein reaches the cells surface to the time when the resultant activated Notch arrives in the nucleus of the neighboring cell. The two cell model is analysed with the following assumed delay (ranges) for equation (4): The matlab code for this analysis can be found here:
 * ~ delay ||~ value ||~ unit ||
 * = T md || [16, 68] || [min] ||
 * = T pd || 5.5 || [min] ||
 * = T exp || 15 || [min] ||
 * = T NA || <<15 || [min] ||

=Results= __To compare the results of the paper with the results obtained through the matlab analysis, click here__.

Direct Autoinhibition of Gene Expression
The results obtained from the analysis of equation (1) are shown in figure 3 in the paper (p. 1401). These are reproduced below: These results match the results in subfigure C, D and E. The attenuation of the protein synthesis in either gene has effects on its period behavior. However, //her7// is more sensitive to the attenuation and turns in to a damped oscillation. This shows that blocking a gene like //her1// is only effective in severing its function if the blocking agent is highly effective (>90%). For //her7// this is clearly less so.
 * ~  ||~ //her1// ||~ //her7// ||
 * ~ a=4.5 || [[image:fig3_her1_normal_normal.png width="480" height="143"]] || [[image:fig3_her7_normal_normal.png width="480" height="143"]] ||
 * ~ a=0.45 || [[image:fig3_her1_normal_at10.png width="480" height="143"]] || [[image:fig3_her7_normal_at10.png width="480" height="143"]] ||
 * ~ a=0.225 || [[image:fig3_her1_normal_at20.png width="480" height="143"]] || [[image:fig3_her7_normal_at20.png width="480" height="143"]] ||

Noisy Control of Gene Expression
The results obtained from the analysis of equation (3) are shown in figure 3 in the paper (p. 1401). These are reproduced below: These results match the results in subfigure C, D and E. The attenuation of the protein synthesis in either gene has much less effect than it did in the deterministic version of the direct autoinhibition model. The stochasticity appears to increase the mRNA synthesis' sensitivity to the fluctuations in concentration and retains a higher number (amplitude) even when the protein synthesis is 20 times attenuated.
 * ~  ||~ //her1// ||~ //her7// ||
 * ~ a=4.5 || [[image:remisysbio/fig3N_her1_normal_normal.png width="480" height="143"]] || [[image:remisysbio/fig3N_her7_normal_normal.png width="480" height="143"]] ||
 * ~ a=0.45 || [[image:remisysbio/fig3N_her1_normal_one_10th.png width="480" height="143"]] || [[image:remisysbio/fig3N_her7_normal_one_10th.png width="480" height="143"]] ||
 * ~ a=0.225 || [[image:remisysbio/fig3N_her1_normal_one_20th.png width="480" height="143"]] || [[image:fig3N_her7_normal_one_20th.png width="480" height="143"]] ||

Delta-Notch Communication in Adjacent Cells
The results from the analysis of the two-cell pathway are shown below, the originals can be found in figure 4 (p. 1404): These results match the results in subfigure B, C and D. In the uncoupled state, the blue and green lines (cell1 and cell2) differ in phase and are not synchronized. With coupling, with T N =36 and 56 minutes, this is fixed. More results can be obtained by using the matlab file.
 * ~  ||~ //deltac// ||
 * ~ no coupling || [[image:fig4_deltaC_normal_nocoupling.png width="480" height="143"]] ||
 * ~ coupling, T N = 36 min || [[image:remisysbio/fig4_deltaC_normal_C36.png width="480" height="143"]] ||
 * ~ coupling, T N = 56 min || [[image:fig4_deltaC_normal_C56.png width="480" height="143"]] ||

=Discussion= Although the paper is ill-structured and chaotically written, it provides an interesting take on the somitogenesis' oscillating behavior. Lewis provides a mathematical approach to solving this question by looking at negative feedback looks and inter cellular communication through the Notch pathway. There are however, a few remaining questions and issues that go left unchecked:
 * Although the negative feedback loops and inter-cellular communication show that temporal oscillatory behavior can be achieved this way, it does not give an explanation on how the oscillations arrest overtime when the oscillatory impulse is removed.
 * The oscillations in the somites are continuous while the somites form in more distinct patterns, this is not explained in the paper.
 * An assumption is made on the amount of negative feedback in equation (1), this relationship (f(p) = k/(1+p 2 /p 0 2 ) ) is not referenced and therefore not very credible on its own.
 * The models period and delay were compared with results found in literature, however the same wasn't done for the concentrations of the mRNA or the protein.
 * The author makes an assumption on the lifetime of protein and mRNA molecules (they are very short), no references are given to support this claim; it seems purely for convenience: '//the behavior is easiest to analyze in the limit in which the lifetimes of the mRNA and protein are very short compared with the total delay time T = Tm + Tp.//'.
 * The behavior of the attenuation examples given in the single cell system without noise provide an interesting lesson for biologists. The power of the oscillatory behavior needs to be taken in to account when designing //in vitro// experiments.
 * The added noise to the single cell system gives remarkable results where the oscillations stay on for much higher attenuation factors. The reason behind this behavior is not explained.
 * Even though the paper produces a lot of different results on several model approaches; the author did not include a detailed parameter estimation on the limit of oscillatory behavior for the several factors. This would have been a welcome results for experimental design purposes.

=Matlab Code= All the matlab code used to reproduce the results in this paper are downloadeble down below, or in pdf format: here.

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