# Sharpening of expression domains induced by transcription and microRNA regulationwithin a spatio-temporal model of mid-hindbrain boundary formation

- Sabrina Hock†
^{1, 2}, - Yen-Kar Ng†
^{1, 3}, - Jan Hasenauer
^{2, 3}, - Dominik Wittmann
^{1, 2}, - Dominik Lutter
^{1}, - Dietrich Trümbach
^{3}, - Wolfgang Wurst
^{3, 4, 5}, - Nilima Prakash
^{3}Email author and - Fabian J Theis
^{1, 2}Email author

**7**:48

https://doi.org/10.1186/1752-0509-7-48

© Hock et al.; licensee BioMed Central Ltd. 2013

**Received: **7 January 2013

**Accepted: **20 June 2013

**Published: **25 June 2013

## Abstract

### Background

The establishment of the mid-hindbrain region in vertebrates is mediated by theisthmic organizer, an embryonic secondary organizer characterized by awell-defined pattern of locally restricted gene expression domains with sharplydelimited boundaries. While the function of the isthmic organizer at themid-hindbrain boundary has been subject to extensive experimental studies, itremains unclear how this well-defined spatial gene expression pattern, which isessential for proper isthmic organizer function, is established during vertebratedevelopment. Because the secreted Wnt1 protein plays a prominent role in isthmicorganizer function, we focused in particular on the refinement of *Wnt1*gene expression in this context.

### Results

We analyzed the dynamics of the corresponding murine gene regulatory network andthe related, diffusive signaling proteins using a macroscopic model for thebiological *two-scale signaling process*. Despite the discontinuity arisingfrom the sharp gene expression domain boundaries, we proved the existence ofunique, positive solutions for the partial differential equation system. Thisenabled the numerically and analytically analysis of the formation and stabilityof the expression pattern. Notably, the calculated expression domain of*Wnt1* has no sharp boundary in contrast to experimental evidence. Wesubsequently propose a post-transcriptional regulatory mechanism for *Wnt1*miRNAs which yields the observed sharp expression domain boundaries. Weestablished a list of candidate miRNAs and confirmed their expression pattern byradioactive in situ hybridization. The miRNA *miR-709* was identified as apotential regulator of *Wnt1* mRNA, which was validated by luciferasesensor assays.

### Conclusion

In summary, our theoretical analysis of the gene expression pattern induction atthe mid-hindbrain boundary revealed the need to extend the model by an additional*Wnt1* regulation. The developed macroscopic model of a two-scaleprocess facilitate the stringent analysis of other morphogen-based patterningprocesses.

### Keywords

Mid-Hindbrain Boundary miRNA Modeling Spatio-Temporal Model Developemental Biology## Background

Patterning phenomena based on the activation of target genes in aconcentration-dependent manner by diffusive signaling molecules, so called morphogens,are of great biological importance [1, 2] as shown, e.g., in Drosophila [3] and mice [4]. Model-based approaches are often used to investigate the formation ofmorphogen gradients and to analyze the sufficiency of the known regulatory mechanisms.Such models do not consider the cells individually but rather as a continuum anddescribe the macroscopic (or homogenized) dynamics of the process. The emergingmacroscopic models are theoretically justified and have already been employed in variousbiological contexts (see, e.g., [5, 6]). An interesting process of such type is the patterning of the neural plate,a precursor tissue, which gives rise to the vertebrate central nervous system (CNS).

Shortly after gastrulation, the neural plate undergoes patterning along theanteroposterior axis into four distinct regions. These regions are the presumptiveforebrain, midbrain, hindbrain and spinal cord. This subdivision relies on awell-defined and locally restricted expression of genes mediating the action of shortand long-range signaling centers, so-called secondary organizers [7]. Of special interest is the isthmic organizer (IsO), the secondary organizerlocated at the boundary between the prospective mid- and hindbrain. The IsO is necessaryand sufficient for the development of the mid-hindbrain region (MHR) [8] and it is characterized by the localized expression of several transcriptionand diffusive signaling molecules.

In the context of the IsO, eight genes are of special interest: *orthodenticlehomologue 2* (*Otx2*) and *gastrulation brain homeobox 2*(*Gbx2*), two transcription factors initially expressed in the anterior andposterior, respectively, part of the developing embryo abutting each other, and whoseexpression boundary determines the position of the future mid-hindbrain boundary (MHB);*fibroblast growth factor 8* (*Fgf8*), which is necessary for thepatterning of the MHR, and *wingless-type MMTV integration site family member 1*(*Wnt1*), required for the maintenance of the IsO, two“morphogens” secreted from the posterior and anterior, respectively, borderof the MHB; and the *Engrailed* (*En1* and *En2*) as well as the*paired box* (*Pax2* and *Pax5*) transcription factors actingdown- or upstream of Fgf8 and Wnt1 and mediating their patterning/maintenance functionat the MHB [8]. *En1* and *En2* are subsumed under the identifier *En*,and *Pax2* and *Pax5* are subsumed under the identifier *Pax* dueto their conserved biological function in mid-/hindbrain boundary (MHB) development. Allof these genes start to be expressed around mouse embryonic day (E) 8.5 of development.Various loss- and gain-of-function experiments demonstrated that at E10.5, these genesare interdependent and their expression patterns are maintained at least until E12.5.These genes build the basis of a gene regulatory network (GRN) that is necessary for thesharpening and subsequent maintenance of the specific IsO gene expression pattern [8]. One crucial aspect of the IsO function is the spatio-temporally defined andlocalized expression of its constituent genes, including the formation of sharpboundaries between the gene expression domains (for a comprehensive review see [8]).

*in situ*hybridization methods). Thesedata are semi-quantitative and capture the level of transcription relative to theminimal and maximal transcription in the same tissue. Based on those experiments the GRNschematically depicted in Figure 1A was inferred [10]. Using artificial thresholds Wittmann et al. [10] constructed the Boolean model shown in Figure 1B.This Boolean model has been shown to be able to generate and robustly maintain theexperimentally observed steady state pattern [10–12]. However, the usage of artificial thresholds results in the loss ofinformation. To exploit all information contained in the data, Wittmann et al. [10] derived a continuous spatio-temporal model using a discrete to continuoustransformation of the Boolean network. Therefore, the discrete Boolean update functionsare replaced by Hill-type functions [10, 13]. The resulting macroscopic model describes the time evolution oftranscription factor activities. As these activities are confined to individual cellsthere is no spatial evolution on this level. Hence the equations can be treated asordinary differential equations (ODEs) for a each point in space. The tissue scalemorphogen gradients and their dynamics are described using diffusion equations. Bothmodels are coupled to account for secretion and uptake of morphogens, the interfacebetween the models, and the full system is shown in Figure 1C. Using this semi-quantitative spatial modeling approach, several interestingaspects of the MHB formation can be assessed, which cannot be studied using Booleanmodels [10].

The class of macroscopic models for two-scale processes provides the basis of thefollowing theoretical and numerical analysis of IsO gene network and signaling proteins.Complementing previous work we address the existence, uniqueness, positivity ofsolutions for the model. We extend previous studies and analyzed the induction of thegene expression patterns at the MHB, especially, we focused on the formation of sharpspatial patterns. Therefore, we introduced theoretical and numerical tools forsemi-quantitative two-scale processes. Using these tools we found that the model has tobe extended by a regulation of *Wnt1* expression to describe the sharp expressionpattern observed *in vivo*. Based on this insight we analyzed, which regulationmechanisms allows for the specific *Wnt1* expression pattern, focusing onpost-transcriptional regulation by miRNAs. MiRNAs are short (∼ 22 nucleotideslong) non-coding RNAs which post-transcriptionally regulate the gene (mRNA) expression [14–16]. This is achieved by binding of the miRNA to the mRNA, repressing thetranslation of the mRNA into protein. Furthermore, if the degree of miRNA-mRNAcomplementarity is high miRNAs induce the decay of the mRNA [17–19]. MiRNAs play a critical role in diseases such as cancer [20] and neuro-degeneration [21] as well as embryonic development [16, 22, 23].

## Results

### Macroscopic, semi-quantitative model of a two-scale process

*Otx2*,

*Gbx2*,

*En*and

*Pax*and the twomorphogen encoding genes

*Fgf8*,

*Wnt1*, which are described in theBackground section, and the morphogens Fgf8 and Wnt1. The interactions areschematically displayed in Figure 1A. The joint dynamicsof transcription factor activities

*u*(

*t*,

*x*) = (

*u*

_{1}(

*t*,

*x*),…,

*u*

_{6}(

*t*,

*x*))

^{T}and morphogen

*v*(

*t*,

*x*) = (

*v*

_{1}(

*t*,

*x*),

*v*

_{2}(

*t*,

*x*))

^{T}are described by an eight-dimensional system of partial differential equations (PDEs)on the spatial domain

*x*∈

*U*= [ −

*L*,

*L*].As the up- and down-regulation of transcription depends only on the concentration oftranscription factors within the individual cells and the local concentration ofmorphogens, the time evolution of the transcription factor activity is governed by areaction equation,

*B*

_{ i }(

*x*,

*u*(

*t*,

*x*),

*v*(

*t*,

*x*)) is potentially anonlinear function of

*x*,

*u*(

*t*,

*x*) and

*v*(

*t*,

*x*).

*B*

_{ i }represents theHill-type regulation of transcription factor

*i*by transcription factors andmorphogens (Figure 1C). The initial conditions for thetranscription factor activity are given by the spatial functions

*u*

_{0i}(

*x*),

*i*= 1,…,6, which describe quantitatively the expressionpattern at E8.5 (Figure 2B upper panels). Hence,

*u*

_{ i }which resembles aconstant translation of mRNA to protein. The dynamics of the morphogen concentrationis then governed by a reaction-diffusion equation,

*d*

_{ j }, denote theconstant synthesis, degradation, and diffusion rate, respectively. In the following,we consider the anterior-posterior direction of the neural tube, which is large incomparison to the MHR and the expected diffusion length scale. Hence, no morphogenarrives at the border, i.e. we have zero morphogen concentration,

*v*

_{ j }(−

*L*,

*t*) =

*v*

_{ j }(

*L*,

*t*) = 0.The initial conditions are given by a two-dimensional vector of spatial functions

*v*

_{j 0}(

*x*),

*j*= 1,2,corresponding to the pattern at E8.5. Hence,

As the PDEs for *v*(*t*,*x*) are linearly coupled with the ODEsfor *u*(*t*,*x*), we can conclude that solutions exist for alltime points *t* ∈ [0,*∞*) [24]. Furthermore, the solution is unique, positive and in$C\left(\right[0,T];{L}^{2}(U;{\mathbb{R}}^{8}\left)\right)$ if the initial condition vector(*u*_{0},*v*_{0})^{T} is positive andbounded [24]. This means that we obtain a solution which is continuous with respect totime and a square integrable function with respect to space and we can consider theasymptotic steady state of the system. Furthermore, the insight that the solutionsare square integrable in *x* ensures the convergence of finite-differencemethods, which are described in “Methods”. In the simulation, specialattention has to be paid to the non-differentiabilities occuring at least for*Otx2* and *Gbx2* in the stationary limit [12].

### MHB model predicts stable steady state patterns

Biologically, one important characteristic of the GRN at the IsO is the activation ofthe genes in a precise spatio-temporal manner and the correct positioning of theirexpression domains [8]. Once the pattern is reached it has to be maintained even if the wholesystem is slightly perturbed for example by transcription noise. This implies thatthe model has to exhibit a stable stationary limit solution, which resembles thesteady state gene expression pattern that is observed at E10.5.

*Otx2*and

*Gbx2*an analytical calculation of the steady state solution is possiblebecause they form a simple genetic toggle-switch system independent of the othercomponents. This system is well studied and we knew that it exhibits separatedexpression domains depending on the interaction strength of the two players [25, 26]. For simplicity and because we lacked detailed knowledge of theinteraction parameters of

*Otx2*and

*Gbx2*, we assumed a symmetricparameter setup, i.e.

with *α*_{1} = *α*_{2} and*β*_{1} = *β*_{2}. Here*u*_{1}(*t*,*x*) denotes the expression of*Otx2* and *u*_{2}(*t*,*x*) denotes theexpression of *Gbx2*. Based on the analytical calculation of the steady statewe found that the parameter values for which we obtained a steady state withseparated expression domains have to satisfy $\frac{{\beta}_{1}}{{\alpha}_{1}}>{(n-1)}^{1+1/n}/n$. Furthermore, the point *x*^{∗}where both expression domains abut each other in the stationary limit is determinedby the initial conditions of *Otx2*, denoted by*u*_{01}(*x*), and *Gbx2*, denoted by*u*_{02}(*x*). If*u*_{01}(*x*)>*u*_{02}(*x*) for*x*<*x*^{∗} and*u*_{01}(*x*)<*u*_{02}(*x*) for*x*>*x*^{∗} then the boundary is placed at*x*^{∗}.

In the following, the MHB is placed in the middle of the considered interval,*x*^{∗} = 0. Furthermore, similar to [10, 12], we set *n* = 2, *k* = 0.1, and*α*_{
i
} = *β*_{
i
} = 1for *i* = 1,…,8. The diffusion coefficients of *Fgf8*and *Wnt1* were reduced compared to previous publications to*d*_{1} = *d*_{2} = 0.001.For the chosen diffusion parameters the values ${v}_{1}(t,-\frac{L}{2})$ and ${v}_{2}(t,\frac{L}{2})$ are approximately zero, ensuring that the Dirichletboundary conditions have no significant influence on the dynamics or the steady stateof the systems, as assumed in the modeling process. For this setup we performed anextensive simulation study to determine the non-trivial steady states of the system(see “Methods” for details). If the initial transcription factor andmorphogen patters are unimodal, we find two stable, non-zero stationary limitsolutions. While the steady state 1 (Figure 2B, leftpanel) shows a maximal width for the expression domain of *En* and*Pax*, for steady state 2 (Figure 2B, rightpanel) this width is minimal. The panels in Figure 2B showexemplary trajectories leading to the steady states. The simulation study indicates,that these are the only steady states reachable from initially unimodal expressiondomains.

For initially multimodal expression domains, we observed more complex expressionpattern, e.g., single spots of *En* or *Pax* expression. These arise as*En* or *Pax* locally exceed the threshold of their mutualactivation, yielding isolated point regions with maximal *En* or *Pax*expression, respectively. The spots are positioned in the region between the minimaland maximal width expression domain given by the steady states mentioned above. Wedid not consider the steady states with expression spots as they are not thought tobe of biological relevance, however this shows that the model is capable of producinga whole spectrum of steady states. Subsequently, we analyzed the stability of the twonon-trivial steady states to understand the temporal dynamics in their closeproximity. We found that both non-trivial steady states are stable. For both steadystates that we consider, the expression of *Otx2*, *Gbx2* and*Wnt1* are the same, whereas for *Fgf8*, *En* and*Pax* we observe a difference in the width of the individual expressiondomains.

### Predicted steady states coincide qualitatively with the experimentalobservations

We compared our steady state solutions and their dependence on initial conditions andparameters to experimentally validated expression patterns. In our model the onlyfactor which determines the position of the MHB is the initial expression pattern of*Otx2* and *Gbx2*. This major role of *Otx2* and*Gbx2* has already been shown in *in vivo* knock-out or knock-inexperiments, in which a change of *Gbx2* and *Otx2* expression domainsalso led to a change in the position of the MHB [27, 28]. Furthermore, both steady states show a restriction of the *En* and*Pax* expression to the MHB region, which can also be observed *invivo*[8]. The steady state with the more restricted initial pattern(Figure 2B right panels and Figure 2C) shows a tighter expression domain. This tighter domain is due to theactivating interaction between *En* and *Pax*, whereas the other steadystate is dominated by the activating interaction of *Fgf8* and *Wnt1*with *En*.

Concerning the expression domain of *Fgf8* we found that in steady state 2 theexpression is restricted in posterior direction due to the regulation of*Fgf8* by *En* and *Pax*. Only in this steady state a sharpboundary of the *Fgf8* expression domain in posterior direction occurs(indicated by the dashed circles in Figure 2B,D). Thisagrees with the experimental observations that the expression of *Fgf8* isrestricted to a ring at the caudal border of the MHB [29]. Furthermore, we concluded that the interactions which are not necessaryto maintain the expression pattern according to [10], are required to sharpen the expression domain of the morphogen*Fgf8*. Those interactions are the activation of *En* by*Fgf8* and *Wnt1* and the activation of *Fgf8* by *En*and *Pax*, which were not considered in [12].

Finally, we considered the expression pattern of *Wnt1*, which is the same inboth steady states. The IsO is a signaling center and one of its main tasks is togenerate a well-defined *Wnt1* gradient. The gradient results from the sharplyrestricted and positioned expression domain of *Wnt1*. However, unfavorablesmooth interfaces of expression domains occur if expression is only regulated by amorphogen in our case *Fgf8*[30]. Depending on the diffusion coefficient of *Fgf8* and theactivation of *Wnt1* with *Fgf8*, the expression domain of*Wnt1* becomes increasingly smooth. This disagreement with the experimentalresults where *Wnt1* is expressed in a narrow ring at the rostral border ofthe MHB with a clearly visible boundary [8, 31–33]. For a detailed illustration and semi-quantitative assessment of theexpression domain of *Wnt1* we refer to the EMAP eMouse Atlas Project [34] (http://www.emouseatlas.org/).

In the following section we will discuss possible post-transcriptional sharpeningmechanisms for the *Wnt1* expression profile. As the more restricted*Fgf8* expression pattern in steady state 2 is closer to the experimentalobservations [29], we will use this steady state in the following analysis, however, theresults are similar if steady state 1 is used. In particular the *Wnt1*expression patterns are alike for both steady states.

### Sharpening of the *Wnt1* expression by miRNA interactions

*Wnt1*expression domain (see Figure 2B), there has to exist an unknown mechanism enforcing the sharpening ofthe

*Wnt1*profile over time, which is not considered in the model. In thiswork we consider post-transcriptional miRNA regulation as recent studies proved thatmiRNAs regulation is particular active at low mRNA copy numbers, which occur in ourmodel simulation at the

*Wnt1*expression domain boundary, and can induce geneexpression thresholds [30, 35]. This might results in a spatial sharpening of the target expressiondomain in the overlapping region [30, 35] via one of the following mechanisms:

Here, *m*(*t*,*x*) is the relative concentration of miRNA,*c*(*t*,*x*) is the relative concentration of mRNA-miRNAcomplex, and *u*_{4} is the relative *Wnt1* mRNA level.*β*_{
m
} is the degradation rate,*d*_{
m
} is the diffusion coefficient of the miRNA and*α*_{
m
}(*x*) is the space dependent productionprofile. Following the suggestions in [30] we used a sigmoidal shaped function*α*_{
m
}(*x*) = *p*_{1}(tanh((*l* − *x*)/*p*_{2}) + *p*_{4})to model the spatial dependence of the miRNA synthesis. The interaction strength ofmiRNA and *Wnt1* mRNA is given by the binding rate *κ*, thedegradation rate of the resulting mRNA-miRNA complex is *λ*, and thedegree of miRNA recycling is denote by *ξ*.

Given this general model, the three scenario can be described by different parametersets. For scenario i), the turn-over parameters of the miRNA are set to zero*α*_{
m
}(*x*)≡0 and*β*_{
m
} = 0, assuming time-independentoverall miRNA level. Furthermore, miRNA is assumed to be completely recycled,*ξ* = 1, but not transported,*d*_{
m
} = 0. In contrast, for scenario ii) andiii) *ξ* = 0,*α*_{
m
}(*x*)≥0∀*x* ∈ *U*>0, and*β*_{
m
}>0. For these to parameters merely thediffusion coefficient is different, i.e. for ii)*d*_{
m
} = 0 and for iii)*d*_{
m
}>0. For all scenarios *κ*>0 and*λ*>0.

For this extended macroscopic model the existence, positivity and uniqueness is alsoguaranteed if and only if *α*_{
m
}(*x*) is abounded, Lipschitz continuous function. This is the case as we assumed*α*_{
m
}(*x*) to be a production profile withvalues in [0,1]. Hence, we could conclude that unique, positive solutions exist forthe extended model and we can simulate the model with the proposed algorithm. It isnot surprising that an overlap of the miRNA and *Wnt1* profile leads to asharpening of the *Wnt1* expression profile. However, we are especiallyinterested in the spatial shape an miRNA expression domain must have to sharpen the*Wnt1* boundary sufficiently as this can easily be compared to experimentalfindings and will lead to predictions for possible regulating miRNAs. In this contextwe defined a sharpening as reduction of *Wnt1* in anterior direction and noreduction at the MHB, i.e. an increase in the second derivative with respect to*x* of the *Wnt1* transcription level for*x* ∈ (0,*L*/2).

We simulated the three scenarios for different parameter sets(*α*_{
m
},*d*_{
m
},*κ*,*λ*)using the same initial conditions and parameters for the original model componentsand the profile function*α*_{
m
}(*x*) = 0.1(tanh((*l* − *x*)/0.3) + 1)with varying *l* (the profile is shown in Figure 3E). A representative simulation result is shown in Figure 3C). In addition, we studied the influence of the miRNA expressiondomain on the sharpening effect in the different scenarios. Therefore we varied theoverlap of the domains by increasing the profile function parameter *l* from 0(no-overlap) to 1 (constant production along the whole MHR). The results are shown inFigure 3D. We found that for scenario i) and ii) themiRNA level and the *Wnt1* expression domain have to overlap in the regionwhere *Wnt1* shows intermediate expression, i.e.*l* ∈ [0.4,0.5]. For both scenarios a complete overlap ofboth domains leads to a overall reduction of *Wnt1* and we see no specificsharpening, especially for scenario i) a complete overlap led to a *Wnt1*knock-down state for the set of considered parameters. For scenario iii) the domainsmust only slightly overlap due to diffusion, if the domains strongly overlap themiRNA diffusing from the posterior direction leads to a blurring of the *Wnt1*domain at the MHB. We conclude that this scenario is not suitable for the MHB modelif we have a strongly overlapping miRNA domain. In the following, we focus onscenario i) for a low and ii) for a high degree of mRNA-miRNA complementarity [17–19].

### miR-709 regulates *Wnt1* mRNA expression in vitro

Up to this point, hypotheses about a possible regulation of *Wnt1* by miRNAswas based on available knowledge about miRNA-mRNA interaction [30, 35] and simulation studies of the MHB model. To gain additional insight, asearch for experimental evidence of miRNAs possibly regulating the *Wnt1* mRNAexpression was conducted. Therefore, we performed a miRNA target prediction andexperimentally validated the predicted miRNAs by determining their expression patternin the developing mouse embryo (with a special focus on the MHR) and their ability totarget the *Wnt1* mRNA (3’UTR) *in vitro*.

We identified potential miRNAs targeting the *Wnt1* mRNA using a combinationof several target prediction tools (see “Methods”). To reduce the numberof false-positive predictions, we post-processed the resulting list using logicfiltering. Therefore, we defined a logical state (ON/OFF) table for the three mouseembryonic stages, namely E8.5, E10.5 and E12.5, including *Wnt1* and the otherfive IsO genes. E12.5 is considered as the IsO gene expression pattern, which hasrefined to its sharp domains at E10.5, is still maintained at E12.5. Assuming that afunctional miRNA targeting *Wnt1* changes the amount of produced Wnt1 proteinand hence the expression state of genes downstream of Wnt1, we concluded that thosegenes change its expression state during miRNA expression. The target prediction wasfiltered for those miRNAs that target at minimum two “OFF genes” and no“ON genes” for each defined developmental stage. This resulted in a listof four miRNAs possibly regulating *Wnt1* mRNA expression (Additional file1: Figure S1). From these possible candidate miRNAs(*miR-705* and *miR-709*) were selected by ranking the interactionsaccording to the prediction scores provided by the distinct prediction tools.

To establish whether these two predicted miRNAs are expressed at the MHB in a patternthat is consistent with the model assumptions, their transcriptional profile in E10.5and E12.5 wild-type mouse embryos was determined. Therefore, we used a very sensitiveradioactive *in situ* hybridization method (for details see“Methods”) to detect the expression profile of these miRNAs along theentire anterior-posterior extent of the MHR on sagittal sections of theseembryos.

*miR-705*is expressed only very weakly across theMHB and in the midbrain and rostralmost hindbrain, whereas

*miR-709*isexpressed strongly and uniformly across the MHB and in the midbrain and rostralhindbrain (see Additional file 2: Figure S2 and Additionalfile 2: Figure S3). Such a profile would lead to an overallreduction of

*Wnt1*mRNA in the model proposed but has no sharpening effect.In the E12.5 mouse embryo, by contrast,

*miR-705*was expressed strongly anduniformly in the entire MHR, including the midbrain, MHB and rostral hindbrain.However, we noted a graded expression of

*miR-709*across the MHB in theventral MHR (which is the region used to determine the expression profiles of theother IsO genes in the considered model) at this stage. Transcription of

*miR-709*at E12.5 was highest in the midbrain, declined towards the MHBand was lowest in the rostral hindbrain, the region of the hindbrain that is underthe influence of the IsO (see Figure 4A). This graded

*miR-709*expression profile became even more evident when we calculatedthe grayscale profile of

*miR-709*expression across the MHR, i.e. from theanterior end of the midbrain to the posterior end of the rostral hindbrain, as shownin Figure 4B (see “Methods”).

Following [30] we used the miRNA profile function*α*_{
m
}(*x*) = *p*_{1}(tanh((*l* − *x*)/*p*_{2}) + *p*_{3})and in order to analyze the sharpening effect of this profile we estimated theparameters*p* = (*p*_{1},*p*_{2},*p*_{3})and *l* (the resulting profile is shown in Figure 3E). We obtained a best-fit profile according to the grayscale profile(Figure 4B red profile). The profile is in accordancewith the modeling assumptions and we assumed that the expression pattern of the IsOgenes is already stably established at E10.5 and this particular miRNA profile is notestablished before E12.5. This is evidence that the miRNA regulation of *Wnt1*in the model is not a regulatory interaction necessary to establish the pattern, butrather acts as a fine tuning mechanism to reduce the range of cells which have anintermediate *Wnt1* expression. To verify this effect the model was simulatedusing the E10.5 expression pattern as initial condition and the estimated profile formiRNA production. We used scenario ii) for the simulation (see Figure 4C and D), because scenario i) mostly affects the repression of*Wnt1* translation and we only have evidence for degradation with theperformed experiments. We also neglect iii) as we had no experimental evidence for adiffusion of *miR-709* in the neural tube. In the simulations, a clearlyvisible sharpening effect could be observed, especially if we increase the*Fgf8* diffusion coefficient (see Figure 4C andD).

In contrast to *miR-709*, did not show a refined and graded expression aroundthe ventral MHB in the E12.5 mouse embryo (see Additional file 3: Figure S3). These results indicated that only *miR-709* isexpressed within the MHR and around the MHB in a pattern as predicted by the modeland consistent with a possible regulatory function of *miR-709* on*Wnt1* expression at the MHB.

To determine whether *miR-709* and *miR-705* can indeed regulate theexpression of *Wnt1* in a cellular context, we used the so-called luciferase“sensor assays”. In this experimental setup, a fragment of the *Wnt13’UTR* containing the predicted *miR-709* and *miR-705*binding sites (BS) was cloned at the 3’ end of a sequence encoding the Fireflyluciferase protein. This constitutes the so-called “sensor vector”. Theluciferase protein transfected with the sensor vector are indirectly measured by abioluminescence reaction resulting in the emission of light, and the intensity of theemitted light is directly proportional to the luciferase protein concentration in thecells. Co-transduction of the cells with the *Wnt1 3’UTR* sensor vectorand *miR-709* or *miR-705* should result in a reduction of luciferaseprotein levels, relative to an “empty” control vector (that does notcontain any *Wnt1 3’UTR* sequences), if and only if these miRNAs bind totheir target sites within the *Wnt1 3’UTR* and post-transcriptionallyinhibit the expression of luciferase protein in these cells. Indeed, co-transductionof HEK-293T cells with the *Wnt1 3’UTR* sensor vector and*miR-709* resulted in an approx. 23% reduction of luciferasebioluminiscence in these cells (see Additional file 3:Figure S2), whereas co-transduction of HEK-293T cells with the *Wnt13’UTR* sensor vector and *miR-705* had no significant effect (seeAdditional file 3: Figure S3). This result indicated that*miR-709*, but not *miR-705*, indeed targets the*Wnt13’UTR*. Next, we determined whether the post-transcriptionalregulation of *Wnt1* expression by *miR-709* is indeed mediated by thepredicted *miR-709* BS within the *Wnt1 3’UTR*. Therefore, thepredicted BS sequences were changed such that they could not be bound anymore by*miR-709* (see Additional file 4: Table S1) andconsequently the expression levels of luciferase protein should not be affectedrelative to the control (“empty”) sensor vector. Mutagenesis of the twopredicted *miR-709* BS within the *Wnt1 3’UTR* in fact abolishedthe negative regulation of luciferase expression from the sensor vector by*miR-709*. This result indicated that the negative post-transcriptionalregulation of the *Wnt1* mRNA by *miR-709* is in fact mediated by thetwo predicted *miR-709* BS within the *Wnt1 3’UTR*.

## Discussion

### Two-scale models allow for the simulation and analysis of complex patterningprocesses including discontinuities

As many biological processes involve different spatial scales, multi-scale modelsbecome increasingly important in computational biology. However, the methodsavailable to simulate and analyze these models rigorously, are still limited. In thiswork, we considered the two-scale processes responsible for the formation of the geneexpression pattern constituting the IsO. This process involves highly nonlineardynamics given by the gene regulatory networks in the single cells, as well as thediffusion of morphogens on the tissue scale. While the dynamics of the tissue scaleare defined by a typical morphogen based patterning process, which has beenextensively studied [2], the single cell system considered here is capable of generatingdiscontinuous expression profiles. Due to numerical diffusion, common numericalmethods fail to converge for this class of models [10]. To circumvent this problem, we used an algorithmic scheme which exploitsthe structure of the model, namely the two-scale nature. It merely uses finitedifferences in the spatial coordinates, which have been successfully applied toreaction-diffusion type models [38], and stiff, adaptive solvers for the time integration. Beyond standardpatterning systems, based on mass-action kinetics, the method applied can solvesystems with highly nonlinear interaction terms. As Lipschitz continuity andboundedness of the activation function is the only prerequisite for the existence andthe boundedness of solution, the results are widely applicable.

The two-scale modeling approach and the tailored simulation scheme are used toanalyze the dynamics of MHB formation and the corresponding steady state expressionpattern. This problem has been approached previously, however, direct numericalintegration using common PDE solvers merely allowed for the study of the short-termresponse [10]. To study the systems in the stationary limit, spectral methods wereemployed [12]. For the model published by Wittmann et al. [10], the spectral method determined four steady states, two of which werebiologically plausible. However, the spectral methods provided only an approximationof the steady state distributions, as they were based on a reduced model and thediscontinuities had to be predefined. This approach indirectly constrained the statespace of the model and the simulation-based stability analysis we carried outrevealed that one of these steady states was unstable, while the other one was stableand corresponds to the steady state shown in Figure 2B inthe left lower panel, with diffusion coefficients set to 0.01. However, the steadystate depicted in Figure 2B in the right lower panel wasnot approximated by spectral methods, illustrating their limitations.

### Spatio-temporal model of MHB formation predicts post-transcriptional miRNAregulation

To determine the plausibility of the existing models, the computed steady stateprofiles were compared to the experimentally observed expression profiles. Whilesimulation results and experimental observations agree qualitatively, the model failsto describe the *Wnt1* profile adequately. The obtained expression domain hada smooth gradient like form which contrasts the experimental findings where theexpression domain is a sharply delimited ring around the MHB [8].

The sharpening of the *Wnt1* profile could be caused by different mechanisms,ranging from additional transcriptional regulation to post-transcriptionalregulation. In this work, we focused on miRNA-mediated regulation as miRNA haveproven to be essential in embryonic patterning processes including morphogens, e.g.in zebrafish [39, 40]. Furthermore, we already established the importance of miRNAs in generalbrain development (unpublished data). However, a role of miRNAs in the formation ofthe MHB and in the refinement of the *Wnt1* expression pattern at thisboundary has not been reported so far.

Using our model, we could verify that different miRNA-mediated regulation mechanismscan cause a sharpening of the *Wnt1* expression domain at the MHB. Thissharpening can be induced by different mechanisms, for which we provided ageneralized mathematical model. In contrast to existing models forpost-transcriptional miRNA regulation we thereby also considered the partialrecycling of the miRNA and did not assume that the mRNA-miRNA complex is degradedinstantaneously, which is biologically often not plausible. Given this theoreticalresult, we performed a problem-tailored miRNA target prediction. Two candidatemiRNAs, *miR-705* and *miR-709*, were identified and analyzedexperimentally. The *in situ* hybridization (detection) experiments showed apromising qualitative expression profile for *miR-709*, which is in line withthe predictions made by the model. However, it did not yield insight into thedetailed interactions or the necessity of *miR-709* for MHB development, whichwould require the establishment of a knock-out, knock-down and/or over-expressionmouse model for *miR-709*.

Beyond the complete verification of the miRNA-based regulation of *Wnt1*expression *in vivo*, another question that arises about the mechanism behindthe formation of the observed gradient of *miR-709* expression across the MHB.Possible mechanisms include the regulation of *miR-709* expression by externalfactors or by a direct feedback regulated by *Fgf8* or other factors involvedin the formation of the MHB. The latter could give rise to positive feedback andfurther sharpening not yet considered in the model.

Apart from a post-transcriptional regulation of *Wnt1* mRNA expression bymiRNAs, additional regulatory mechanisms might also influence the formation of sharp*Wnt1* expression boundaries at the MHB. An example is the transcriptionfactor *Lmx1b*, which is known to maintain *Wnt1* expression [41, 42] at the MHB at E10.5. As this factor is expressed only in a ring around theMHB, it might contribute to the sharpening of the domain. Other signaling andadditional cell-cell communication processes can also not be ruled out. Additionalstudies are necessary to unravel or exclude further mechanisms involved in the finetuning of the IsO gene expression profiles during mouse (vertebrate) embryonicdevelopment.

## Conclusion

To understand complex patterning processes quantitatively, spatial-temporal modeling hasbeen proven to be essential, for example in Drosophila. In this contribution, weillustrated how even a model using only a semi-quantitative description of the generegulatory network acting on a tissue scale can help to guide the discovery of novelregulation mechanisms, in our case gene expression boundary sharpening induced bypost-transcriptional feedback mechanisms. As spatially resolved data increases quickly,methods employing also spatial information will become more and more important.

## Methods

### Numerical integration

_{ x }denotes a discretizedLaplace operator we obtained an initial value problem for ODEs given by

for each grid point *x*_{
i
},*i* = 1,…*N*. As all solutions are in$C\left(\right[0,T],{L}^{2}([-L,L],{\mathbb{R}}^{8}\left)\right)$ we needed a *L*^{2} stable spatialdiscretization and we chose finite differences. We took into account that with*h*→0, where *h* is the grid width, the generated ODEs becameincreasingly stiff. The finite differences method yields a Jacobian matrix with aspecial sparsity pattern. The matrix is non zero only on the eight subdiagonals andthe eight superdiagonals, which we use to enhance the performance of the ODE solver.The algorithm is implemented for MATLAB R2012a and can befound in Additional file 5: Data S1.

### Steady state approximation and stability analysis

We determined the steady states by simulating the model with the parameters given inthe results section from different initial conditions until the calculated state ofthe system changed less than machine accuracy between two time steps. This wasrepeated for many different initial conditions, space fillingly sampled, to find asmany steady states as possible with a numerical simulation.

In a second step, we identified the stability of the reached state. Therefore, weadded uniform distributed noise, $\u03f5\sim \mathcal{U}\left(\right[-0.001;0.001\left]\right)$, to the calculated value for each component under theconstraint that*u*_{
i
}(*t*,*x*) + *ϵ*≥0for *i* = 1,…,6 and*v*_{
j
}(*t*,*x*) + *ϵ*≥0for *j* = 1,2. We used the obtained value as the initial conditionfor the approximation of the steady state. If the unperturbed state was reached againwe concluded that the state was stable.

### Experimental animals

Outbred CD-1 mice were purchased from Charles River (Kisslegg, Germany) and keptunder a 12-12 light-dark cycle under standard conditions. Mice had ad libitum accessto food and water. Noon on the day of vaginal plug detection was designated asembryonic day (E) 0.5. Embryos were staged according to Theiler [43].

### miRNA prediction

To improve the robustness of the predicted miRNAs that target the *Wnt1* mRNA,data sets of five most commonly used miRNA prediction tools were used in combination.A miRNA target was considered as a candidate if the miRNA target interaction waspredicted by at least three out of five miRNA target prediction tools. For the miRNAtarget prediction, we used the following publicly available tools: TargetScan [44], PicTar [45], miRNAMAP (miranda) [46], TargetSpy [47] and miRBase (DIANA) [48].

### Radioactive *in situ* hybridization (ISH) and probe labeling

Timed-pregnant mice were killed by *C* *O*_{2} asphyxiation.Uterine horns were removed and kept in cold phosphate buffered saline (PBS) beforedissection of the embryos. Embryos were fixed in 4% paraformaldehyde (PFA) (Sigma,Germany) in PBS overnight, dehydrated in an ascending ethanol series, cleared inxylene, embedded in paraffin, and sectioned on a microtome (Microm, Germany) at 8*μ* *m* thickness. Radioactive locked nucleic acid (LNA)-basedISH using unlabeled, LNA-modified *mmu-miR-709* (Exiqon, Denmark, Cat No39324-00) and *mmu-miR-705* (Exiqon, Cat No 39320-00) detection probes wereperformed using an ISH protocol as described in [33] with minor modifications: the proteinase K treatment was omitted,pre-hybridization and hybridization of the labeled probes was done in an *insitu* Hybridization Buffer (Ambion, USA, Cat No B8807G) at 53°C, andpost-hybridization washes were done sequentially in 1xSSC, 0.2xSSC and 0.1xSSC at53° C. Sections were counterstained with Cresyl Violet (0.5%, Sigma) accordingto standard procedures after exposure for 1–3 weeks. Images were taken with anAxioplan2 microscope using bright- and darkfield optics, AxioCam MRc camera andAxiovision 4.6 software (Zeiss, Germany), and processed with Adobe Photoshop CS5software (Adobe Systems Inc., USA). The LNA-modified *mmu-miR-709* and*mmu-miR-705* detection probes were labeled with [*α*^{35}*S*]-dATP (GE Healthcare, USA), using theTerminal Transferase Labeling Kit (Roche, Germany) according to themanufacturer’s instructions, with minor modifications: a 1:50 dilution (0.5*μ* *M*) of the unlabeled LNA-modified detection probe, 1*m* *C* *i*/*m* *l* *α*^{35}*S*-dATPand no UTP were used in the reaction mixture.

### Calculation of grayscale profile and profile fitting

We defined a 300 pixel long and 15 pixel wide region from the approximate anteriorend the midbrain to the approximate posterior end of the rostral hindbrain (bothmarked by dashed red lines in Figure 4A) on a darkfieldpicture taken from a sagittal sections of an E12.5 wild-type embryo hybridized withthe radioactive *mmu-miR-709* detection probe). Using the software ImageJ(NIH, USA), we calculated the gray value in this picture at each pixel within therectangular area in Figure 4A and averaged the valuesalong the width of the rectangular. The gray value profile obtained was normalizedagainst the mean gray value intensity in the two light blue squares/boxes shown inFigure 4A. We estimated the parameters*p* = (*p*_{1},*p*_{2},*p*_{4})and *l* of the profile function*α*_{
m
}(*x*) = *p*_{1}(tanh((*l* − *x*)/*p*_{2}) + *p*_{4})(suggested in [30]). Therefore, we minimized the quadratic distance $\sum _{i=1}^{300}\left(D\right({x}_{i})-{\alpha}_{m}({x}_{i},p,l\left)\right){)}^{2}$, with pixels *x*_{
i
} and grayvalue *D*(*x*_{
i
}), using the minimization methodfminsearch implemented in MATLAB R2012a. The optimalparameters are *p*_{1} = 0.3062,*l* = 0.451, *p*_{2} = 0.2,*p*_{3} = 0.064 and*p*_{4} = 2.2868 and the least squares fit of*α*_{
m
}(*x*) to the gray value curve*D*(*x*_{
i
}) is shown in in Figure 3E.

### Luciferase sensor assays

A 857 bp fragment of the *Wnt1 3’UTR* (Entrez Gene Acc. No. NM_021279,basepairs 1496-2352) was amplified from the E12.5 mouse embryo head cDNA using theprimer pair shown in Additional file 4: Table S1. This*Wnt1 3’UTR* fragment contains two putative BS each for*mmu-miR-709* and for *mmu-miR-705* as predicted by miRBase(microCosm). This fragment was subsequently subcloned into the *Xba* I sitelocated downstream of the firefly luciferase stop codon in the pGL3 Promoter vector(Promega, USA). Site-directed mutagenesis of the predicted *mmu-miR-709* seedsequences within the 857 bp *Wnt1 3’UTR* fragment was done using theQuick Change Multi-Site Directed Mutagenesis Kit (Stratagene, USA) according to themanufacturer’s instructions. Mutagenic primers used are shown in Table S4.HEK293T (1×10^{5} cells/well) were plated in a 24-well plate andco-transfected 24 hours later with 300 ng of*Wnt1 3’UTR-WT* or *Wnt13’UTR-MUT* sensor vector, 30 ng of renilla luciferase vector, and*mmu-miR-709* (Ambion, Cat No PM11496) or *mmu-miR-705* (Ambion, CatNo PM11392) precursor miRNA as indicated in the figures, using Lipofectamine 2000(Invitrogen) according to the manufacturer’s instructions. Luciferase activitywas measured 48 hours after transfection using the Dual-Luciferase Reporter AssaySystem (Promega). The firefly luciferase values were normalized against the renillaluciferase values as internal transfection control. As we also observed adown-regulation of luciferase activity after co-transfection of the precursor miRNAand the pGL3 Promoter vector (without any 3’UTR cloned into it) in someinstances, which we considered to be “off-target” effects of thecorresponding miRNA, we always used the co-transfection of pGL3 Promoter vectorwithout 3’UTR (“empty vector”) and the corresponding miRNA as thecontrol in our experiments, and this value was set as one. Transfections were done intriplicate and all data derive from three independent experiments.

### Ethics statement

Animal treatment was conducted under federal guidelines for the use and care oflaboratory animals and was approved by the HMGU Institutional Animal Care and UseCommittee.

## Notes

## Declarations

### Acknowledgements

We thank Susanne Laaß for excellent technical support with the experiments,Christiane Fuchs for useful discussions and critical reading of the manuscript, andDiana Otero for proof reading the manuscript. Furthermore, we would like to thank theunknown reviewers, who provided excellent comments and held to improve the papersignificantly. This work was supported by the Helmholtz Alliance on Systems Biologyproject ‘CoReNe’, the European Union within the ERC grant‘LatentCauses’, the Bayerischer Forschungsverbund Stammzellen(ForNeuroCell Bayern; F2-F2412.18 - 10c/10 086), the BMBF grants ‘Neurogeneseaus Gehirn- und Hautzellen’ (01GN1009C; TP2 u. TP4) and ‘VirtualLiver’ (grant-nr. 315752), and the EU grant ‘Systems Biology of StemCells and Reprogramming’ (SyBoSS [FP7-Health-F4-2010-242129]).

## Authors’ Affiliations

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