diffusion-based physical theories of morphogen propagation

Seminar – 4. letnik

Diffusion-based physical theories

of morphogen propagation

Author: Jurij Sablić

Advisor: prof. dr. Primož Ziherl

Ljubljana, February 2012

Abstract:

The morphogen gradient theory is the most widely accepted model describing how cells in

early embryos acquire their positional information. In this seminar we present the diffusion-

based physical models of Bicoid morphogen gradient establishment that explain the

establishment of hunchback gene domain in early Drosophila embryo.

2 SABLIĆ, Jurij. Diffusion-based physical processes of morphogen propagation. Seminar. Ljubljana, FMF, 22. 2. 2012

Contents 1. INTRODUCTION ............................................................................................................................... 3

2. SINGLE GRADIENT MODEL............................................................................................................... 3

3. BI-GRADIENT MODELS ..................................................................................................................... 7

3.1. Simple bi-gradient model ............................................................................................................. 7

3.2. Activator-corepressor model ....................................................................................................... 8

3.3. Bcd-Staufen model ....................................................................................................................... 9

4. EXPERIMENT .................................................................................................................................. 12

5. CONCLUSION ................................................................................................................................. 14

6. BIBLIOGRAPHY ............................................................................................................................... 15


1. INTRODUCTION

An essential feature during the development of an organism is cell differentiation [1].

Different cells make up different organs and are therefore found at different positions within

an organism. It is of a great importance to know why certain cells start to gather at a certain

position in the organism. It is assumed that in every organism the process of cell gathering

starts in an early phase of its embryonic development. Scientists have been trying for nearly

sixty years to understand the processes that trigger the above described domain specification

in an early phase of organism’s embryonic development. Domain is to be understood as an

embryo region where a specific gene or specific cells appear that are not present at any other

part of the embryo [1].

One of the most promising explanations of the precise domain specification is the

theory of morphogen gradient [1]. Morphogens are proteins that are produced at one end of

the embryo by maternal mRNA and then somehow propagate all over the embryo. Each cell is

believed to have its surface receptors for a specific morphogen and will orient itself towards a

specific position inside the embryo, where the concentration of the morphogen to which the

cell reacts has exceeded the threshold value of cell’s receptors. In this seminar we will try to

explain the physics of morphogen propagation processes, focusing on the development of the

hunchback gene (Hb) along the anterior-posterior axis of Drosophila embryo. This gene is

believed to be induced mainly by Bicoid morphogen (Bcd) [1]. There are several different

theories that explain Bcd concentration gradient formation and the establishment of Hb. We

will first explain a model of diffusion and degradation of one morphogen and show its

weaknesses. Then we shall focus on bi-gradient models, starting with the simple one,

describing diffusion and degradation of both of them, and continuing with the more complex

ones, proposing binding of both morphogens into a complex that also diffuses and degrades.

We will conclude with an experiment and compare the theoretical predictions with

experimental results.

2. SINGLE GRADIENT MODEL

The first and the simplest model that explains Hb formation is the single gradient

model. Bcd is produced at the anterior pole of the embryo by its maternal RNA and then

diffuses along the anterior-posterior axis. We also assume that Bcd is being degraded. The


process of Bcd gradient establishment and gene recognition is schematically shown in Fig. 1.

The process is described by a modified diffusion equation [3]:

(1)

Here B is the concentration of Bcd, D is the diffusion coefficient and ω is the degradation

rate. Both D and ω are constant. The orientation of x-axis is from the anterior towards the

posterior pole. The boundary conditions are:

|

and

|

(2)

J is the production rate of Bcd at the anterior pole and L is the length of the embryo. If we

focus only on stationary state, Eq. (1) reduces to:

(3)

where the decay length √ ⁄ . Its solution is:

(

) (4)

and ( ⁄ )⁄ is the minimum value of Bcd concentration in the embryo [4]. We

can now define the threshold value of Bcd concentration at which the production of Hb is

triggered and calculate the position in an embryo at which this occurs:

( ) (

) (5)

Equation (5) contains the inverse function of hyperbolic cosine, whose argument must be

greater than 1. From that we can calculate the minimum length of the embryo for a specific

threshold value of morphogen concentration as ( ⁄ ). When

hyperbolic sine and cosine in Eq. (4) both have very large arguments and can therefore be

approximated with exponential functions. Thus the concentration profile becomes purely

exponential and can be approximated with ( ⁄ ). The other limit

would result in which would imply that Hb domain spreads along the whole embryo.

However this is not the case. Therefore the latter limit is useless for the description of Hb

domain establishment.


Figure 1: Morphogen gradient formation and gene domain establishment by a single morphogen

diffusion. (A) Morphogen, i.e. Bcd, (green dots) is released from restricted anterior part (green cells).

(A1) One dimensional sketch. (A2) Sketch of planar gradient diffusion. (A3) Graph of morphogen

concentration versus distance from morphogen source. Different target genes are expressed above

different concentration threshold. Above C1 target gene x (red) is activated and above C2 target gene

y (pink) is avtivated. (B) Diffusion model. Morphogen (green) secreted from the source (green cells)

diffuses through the extracellular space (grey grid). As it moves away from the source it generates a

concentration gradient. If the target gene cell receptors (red) recognize the sufficient concentration,

they bind to the morphogen thus forming gene domains [2].

The single morphogen model however suffers from two important weaknesses:

precision and scaling [3]. Precision here is to be understood as a high degree of similarity

between all embryos and scaling as the proportionality between gene expression domains and

embryo size. All grown Drosophilae have approximately the same body architecture,


therefore it is important that all early embryos have approximately the same Hb domain. In

other words, Hb domain in Drosophila embryo is to be precise. Furthermore, the size of the

domain is proportional to Drosophila length. An early embryo has to exhibit the same

characteristic. The embryo is perfectly scaled if the fraction ⁄ is a constant number

between 0 and 1 [4].

Experiments have shown that Bcd gradient is not robust and shows high embryo-to-

embryo variability [3]. Since its threshold value for Hb production remains the same, the

relative position of Hb gene appearance ( ⁄ ) should also display high embryo-to-embryo

variability, according to the single gradient model. It, however, does not.

Moreover, if we consider the functional form ( ) in Eq. (5), we see that it is not

defined for [4]. At its value is and as increases decreases

and asymptotically approaches . Fig. 2 shows the variation of ⁄ against for three

different values of the threshold value of Bcd. As mentioned above, perfect scaling would

correspond to ⁄ , and a complete lack of it to ⁄ ⁄ . Figure 2 clearly

shows that ⁄ curves are everywhere close to ⁄ and nowhere close to constant, which

implies that the single morphogen model is neither precise enough nor sufficiently scaled.

Figure 2: The single gradient model is unable to satisfy the scaling condition. Shown here is the

dependence of normalised position along anterior-posterior axis where Bcd drops below threshold

versus the embryo length (figure adapted from Ref. [4]). Solid lines are the analytical expressions (5)

for three different Bcd threshold concentration values, 0.01 (black), 0.1 (blue) and 0.7 (green). The

dashed lines are ⁄ curves [4].


3. BI-GRADIENT MODELS

In previous section we explained the principles of the single gradient model, the results

it yields and its shortcomings compared to the experimental results. The most discussed

correction of the single gradient model is addition of another morphogen that somehow

influences the robust Hb gene establishment. Various researchers have proposed different bi-

gradient descriptions, and in this section we will present some of them.

3.1. Simple bi-gradient model

This model assumes that the second morphogen, often referred to as Nanos, is produced

at the posterior pole of the embryo from a localized source of mRNA and then diffuses along

the anterior-posterior axis like the first morphogen [4]. Furthermore, it also assumes that the

second morphogen acts as an inhibitor of the first one, allowing Hb activation only in regions

where its concentration is exceeded by the concentration of the Bcd. The morphogens do not

interact chemically. They both obey Eq. (1) or its stationary form, Eq. (3). It is reasonable to

assume that each morphogen has its own decay length. Boundary conditions for the second

morphogen are the opposite of those for Bcd, meaning that the production rate of the second

morphogen, i.e. the derivative of its concentration equals 0 at the anterior pole and at the

posterior pole. The solutions of the two equations are:

(

) and (

), (6)

where stands for the concentration of Bcd and for the concentration of the posterior

morphogen. Constants and equal to ( ⁄ )⁄ and ( ⁄ )⁄ ,

respectively. For embryos much longer than and Eqs. (6) simplify into:

(

) and (

). (7)

Obeying the second assumption of this model, the threshold position can be calculated by

solving the equation ( ) ( ). For sufficiently long embryos this yields:


[ (

)] (8)

However, Eq. (8) generally does not solve the scaling problem, as the relative threshold

position ⁄ is not independent of unless and . In this case Eq. (8)

simplifies to ⁄ , which fulfils the scaling condition. This is valid for any embryo

length, which is clearly visible from Eqs. (6). Furthermore, the latter case is also precise, since

the threshold position for Hb production does not depend on the threshold value of any of the

morphogens that can therefore appear non-robust without affecting the robustness of the

hunchback gene. The described special case of this model is free of the shortcomings of the

single gradient model. The comparison of the two models in discerning the Hb threshold

boundary is shown in Fig. 3.

Figure 3: Setting the Hb boundary in the simple bi-gradient model (circles) and in the single gradient

model (rectangles) for three different values of the decay length . For the former, Hb threshold value

remains constant but for the latter it changes with [3].

3.2. Activator-corepressor model

Activator-corepressor model also proposes two morphogens [5]. The activator is Bcd

protein produced at rate at the anterior end of the embryo, and the other morphogene, i.e.

the corepressor protein, is believed to inhibit Bcd’s action in switching on target genes. The

latter one is produced at rate at posterior end. At each instant the production is

concentrated at a randomly chosen location between and from the respective poles.

The proteins diffuse away from the poles with the same diffusion constant, and are degraded

at the same rate. Bcd and the corepressor bind with a certain rate to form a complex that


diffuses and is degraded in the same way as Bcd and the corepressor. The model assumes that

unbinding process occurs at a far slower rate and can therefore be neglected. Only active Bcd,

i.e. Bcd unbound to the corepressor, can activate Hb transcription by binding to its control

region on the DNA. Hunchback gene is produced from its mRNA. The latter also diffuses and

is spontaneously degraded, but it has different diffusion and degradation constants compared

to Bcd, the corepressor or their complex. Hb protein is therefore synthesised at a rate

proportional to the mRNA concentration and undergoes diffusion and degradation,

characterized by their unique diffusion constant and degradation rate.

These processes are mathematically described by a set of five partial differential

equations that cannot be solved analytically [5]. Numerical solution of the above model for a

realistic set of model parameters corresponding to wild-type embryos is shown in Fig. 4. The

density of Bcd plus the Bcd-corepressor complex decays exponentially away from the anterior

pole (Fig. 4b). The total Bcd profile shows considerable embryo-to-embryo variation, with the

decay length √ ⁄ (Fig. 4e) and the position where the density crosses a threshold value

(Fig. 4f). They both exhibit large fluctuations. However, the profiles of the corepressor and

unbound Bcd fluctuate far less from embryo to embryo (Fig. 4c). Their unbound densities are

very low at mid-embryo which is a consequence of them binding together. The active Bcd is

then able to precisely activate Hb. Hb mRNA and Hb protein densities (Fig. 4d) both pass

their half-maximum values at with little embryo-to-embryo variability,

( ) where denotes embryo length (Fig. 4f). is declared to be Hb

boundary location that scales with , unlike the Bcd profile (Fig. 4g).

3.3. Bcd-Staufen model

This model, as well as all others in Sec. 3, assumes two morphogens, Bcd and Staufen

[6]. The main difference between the simple bi-gradient model, activator-corepressor model

and Bcd-Staufen model is that the latter does not emerge only from posterior end but from

both. Furthermore, Staufen protein reversibly forms a complex with Hb mRNA. In the

anterior part of the embryo, hunchback production is higher due to higher concentration of

Bcd protein. The bonding of Hb mRNA with Staufen balances asymmetric production of the

hunchback gene and enables scaling in the middle of the embryo. The effective transport of

Hb mRNA is self-regulated by the fact that the ensuing changes in the Hb mRNA profile lead


Figure 4: (a) Sketch of Drosophila embryo showing localized Bcd/corepressor sources at opposite

poles, and corepressor-Bcd binding. (b)–(g) Positional information for Bcd and Hb density profiles;

data from 100 simulated embryos. Maximum densities normalized to unity. Mean density profile and

standard deviation for: (b) Total Bcd (Bcd plus complex); (c) Unbound Bcd and the corepressor; (d)

Hb; in (b)–(d), 10 typical individual density profiles are also shown. (e) Distribution of decay length

describing exponential decay of total Bcd density profile. (f) Distribution of positions and

where each Bcd and Hb density profile crosses given threshold, 0.17 for total Bcd, 0.5 for Hb. (g)

Positions and as a function of EL; Bcd: black circles, Hb: red squares. Green straight line

represents [5].

to changes in diffusive flux, which counteract the transport via Staufen. This enables an

effective control of the extent of Staufen assisted transport and thus yields precision and

robustness of the hunchback boundary.

The mathematical description of Bcd-Staufen model is a set of six diffusion and

reaction-kinetics equations that include [6]:

diffusion and degradation processes of Bcd, Staufen and Hb mRNA,

production of Bcd by translation,


formation, disintegration and diffusion of Staufen-Hb mRNA complex,

dynamics of Staufen protein released from anterior and posterior end,

transcription of Bcd into hb mRNA and

transcription of hb mRNA into Hb gene.

Aegerter-Wilmsen and co-workers reported a numerical solution of the equation system

for wild-type embryos [6]. They first solved Bcd diffusion-production-degradation equation

varying the diffusion constant between and ⁄ and embryo length between

and independently. The profile of Bcd was then used as input data in solving the

other equations. The numerical results are shown in Fig. 5. It is clear that although Bcd is not

robust Hb shows a high degree of robustness, meaning that although Bcd profiles show high

embryo-to-embryo variability, Hb boundary appears at approximately the same relative

position in all embryos. This position is at ( ) according to this model.

Figure 5: Numerical results of Bcd-Staufen model obtained by independent variation of Bcd diffusion

constant and embryo length. (a) A set of 100 Bcd profiles that were thus obtained. (c) Hb profiles that

were calculated solving the rest of diffusion and reaction-kinetics equations with the Bcd profile set

shown in (a) used as the input data. Position at which the Bcd gradient (b) and the Hb profile (d) cross

a threshold concentration of 0.19 and 0.5, respectively, versus embryo length [6].


4. EXPERIMENT

In the above sections we presented some theoretical models of Hunchback profile

establishment via morphoges. Now we will present an experiment that tested these models, its

protocols and its results. The experiment was done by Houchmandzadeh and collaborators

[7]. The embryos were cultivated at 25 °C and immunostained by immunofluorescent anti-

Bcd and anti-Hb molecules. The prepared and labelled embryos were then observed under

confocal fluorescence microscope, which enabled high resolution imaging [8]. A typical

image obtained that way is shown in Fig. 6.

Figure 6: Typical image of Bcd staining [7].

Protein profiles were measured in the fourteenth cycle of cell division. Bcd protein

profiles in about 100 wild-type embryos are shown in Fig. 7a. The profile displays a high

embryo-to-embryo variability. To quantify this variability, the position ( ) along the

embryo at which each curve crosses the concentration of 0.23 was measured. These positions

are spread over 30% of the embryo length (EL) (Fig. 7b), and have a standard deviation of

0.07 EL. This means that the positional error of the Bcd gradient is greater than five nuclei in

50% of embryos. Another way to quantify the variability, which is not sensitive to the

normalization of the Bcd protein profile, is to measure the decay length (Fig. 7c). The

standard deviation of is 0.045 EL, which corresponds to the same variability that was

measured for .

Unlike Bcd, Hb protein profile displays an extreme reproducibility from embryo to

embryo. The Hb profile in about 100 embryos from early to late cycle 14 is shown in Fig. 7d.

The Hb distribution was quantified using the point ( ) at which each profile crosses the 0.5

threshold (Figure 7b). The standard deviation of is 0.01 EL, meaning that two-thirds of

embryos have Hb boundaries defined more precisely than the size of one nucleus. The

information about the embryo scale is also revealed at Hb expression level.

Bcd exponential profile appears not to be affected by embryo length. When is

plotted against EL (Fig. 7e), the correlation coefficient is indeed negligible. A similar lack of


correlation is observed between the values of and EL. In contrast to Bcd, the position of the

Hb boundary displays a strong correlation with the embryo length (Fig. 7f).

Figure 7: Positional information of Bcd and Hb gradients. (a) Bcd gradient in about 100 embryos. (b)

Distribution of positions at which each gradient crosses a given threshold: 0.23 for Bcd and 0.5 for Hb.

(c) Distribution of slope of decay length for each Bcd profile. (d) Hb gradient in about 100 embryos.

(e) Position at which each gradient crosses a given threshold versus embryo length (EL) for Bcd. (f)

Position at which each gradient crosses a given threshold versus embryo length (EL) for Hb [7].

Hb mRNA displays the same precision and conservation of proportions as Hb protein.

The spatial position of the hb mRNA boundary in early cycle 14 embryos has a standard

deviation of 0.01 EL, and displays a high correlation with the egg length. The Hb domain

boundary for wild type embryo was measured to be ( ) .

Throughout Sec. 3 we emphasized the importance of Hb domain scaling. As a proof of

that we cited Hb threshold positions yielded by the three described models. In Table 1, we

compare the theoretically calculated values of and the value of obtained experimentally

that we described in Sec. 4.

The results obtained by the three bi-gradient theories exhibit remarkable resemblance

to experimental data. The activator-corepressor and Bcd-Staufen theories give the same result


as the experiment, while the hunchback boundary obtained by the simple bi-gradient theory is

within the range of experimental error. Hunchback gene is thus present in the anterior part of

the embryo and its domain vanishes at approximately half of its length.

Model/experiment

Simple bi-gradient [3] -

Activator-corepressor [5]

Bcd-Staufen [6]

Experiment [7]

Table 1: Hb boundary and its variance obtained by bi-gradient theories and measured in wild-type

embryos.

5. CONCLUSION

In the beginning of this seminar we presented the single gradient model and its

shortcomings, which are precision and scaling. Then we presented three fundamentally

different approaches within the theory of bi-gradient models that successfully explain

precision and scaling. The first was the rather primitive simple bi-gradient model and the

other two were more complex ones, the activator-corepressor model and the Bcd-Staufen

model. We also discussed an experiment where some of the theoretically calculated properties

were measured, and we compared theoretical and experimental results.

The diffusion-degradation models of one or two morphogens fail to explain Hb

domain establishment in Drosophila embryo as the former is neither precise nor scaled, and

the latter yields results that are not within the error bars of the experiment. Only when the

complex reaction kinetics is included in the diffusion-degradation bi-gradient model the

process is described accurately enough.

In this seminar we have discussed only the extracellular diffusion-based theories of

morphogen propagation in one dimension, i.e. anterior-posterior axis. There have however

appeared some indications that extracellular diffusion does not suffice when trying to describe

planar propagation of Decapentaplegic (Dpp) in Drosophila wing disc [9]. Therefore, some

attempts have been made in coupling endocytosis with extracellular diffusion processes [9].


The most promising one is transcytosis, where morphogens undergo repeated rounds of

internalization into cells and recycling [10].

In conclusion, Drosophila embryo and its domain specification may seem as a purely

biological topic. They however are not. We hope we have managed to show that there are

complicated physical processes in the background of what seems to be biology that are still

not sufficiently well explained and therefore remain extensively studied by biophysicists.

6. BIBLIOGRAPHY

[1] T. Bollenbach, K. Kruse, P. Pantazis, M. González-Gaitán and F. Jülicher, Phys. Rev.

Lett. 94, 018103 (2005).

[2] E. V. Entchev and M.A. González-Gaitán, Traffic 3, 98 (2002).

[3] B. Houchmandzadeh, E. Wieschaus and S. Leibler, Phys. Rev. E 72, 061920 (2005).

[4] P. McHale, W.-J. Rappel and H. Levine, Phys. Biol. 3, 107 (2006).

[5] M. Howard and P. Rein ten Wolde, Phys. Rev. Lett. 95, 208103 (2005).

[6] T. Aegerter-Wilmsen, C. M. Aegerter and T. Bisseling, J. Theor. Biol. 234, 13 (2005).

[7] B. Houchmandzadeh, E. Wieschaus and S. Leibler, Nature, 415, 798 (2002).

[8] D. B. Roberts, Drosophila: a practical approach (Oxford University Press, Oxford, 1998).

[9] K. Kruse, P. Pantazis, T. Bollenbach, F. Jülicher and M. González-Gaitán, Development

113, 4834 (2004).

[10] T. Bollenbach, K. Kruse, P. Pantazis, M. González-Gaitán and F. Jülicher, Phys. Rev. E

75, 011901 (2007).

diffusion-based physical theories of morphogen propagation

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