Branch Mode Selection during Early Lung Development
[['Branched structures are ubiquitous in nature , and the mechanism of their formation has been investigated for decades both in experimental [1]–[3] and theoretical studies [4] , [5] .', 'Studies some 50 years ago showed that the dimension of the airway in adult lungs depends exponentially on the branch order very well up to the 10th generation [6]; based on subsequent analysis lungs were suggested to be fractals , with fractal dimension close to 3 for the adult human lung [7]–[14] .', 'Various algorithms that generate lung trees with morphometric characteristics similar to adult mammalian lungs have been reported [4] , [5] , [15] , [16] .', 'In a geometrically realistic model Kitaoka and co-workers required nine basic and four complementary rules to fill the 3-dimensional thoracic cavity with a branching model [16] .', 'The rules that they defined were to a large part similar to those that were later discovered in a careful experimental study of the growing lung [17] .', 'In this later experimental study it was shown that the branching process during lung development is remarkably stereotyped and that the branched tree is generated by the sequential use of three geometrically simple modes of branching , i . e . domain branching , planar bifurcation , and orthogonal bifurcation [17] .', 'Errors like branch displacement are observed in less than 1% of all branching events [17] .', 'Branching is thus not a stochastic process , but must be controlled .', 'This raises the question of how the information required to generate a structure of such complexity is encoded in the genome .', 'Since branched structures are created by recursive processes , a limited number of proteins can , in principle , control all of the branching in the lung from the trachea to the terminal bronchioles [3] .', 'Genetic studies have led to the identification of key regulatory genes and morphogenes that control lung development , most importantly Fibroplast growth factor ( FGF ) 10 , Sonic hedgehog ( SHH ) , and Bone morphogen protein ( BMP ) 4 [1] , [2] , [18]–[23] ( Figure 1a ) .', 'FGF10 is expressed in the distal mesenchyme around the epithelial bud tip and is essential for the lung bud formation , proliferation of the endoderm , and directional outgrowth [3] , [24] , [25] .', 'FGF10 signals mainly through the epithelial FGF receptor FGFR2 , and inactivation of FGFR2 in the lung epithelium results in the disruption of lobes and small epithelial outgrowths that arise arbitrarily along the main bronchi [25] .', 'FGF induces the expression of SHH , BMPs , and Sprouty which in turn limit the expression and signalling of FGFs [24] , [26]–[29] .', 'SHH signals through its receptor Patched ( Ptc ) , and affects FGF10 and BMP4 activity .', 'While many studies show that BMPs regulate lung development their detailed effects have been difficult to disentangle [1] .', 'Adding BMP4 to organ cultures of the whole embryonic lung promotes branching morphogenesis and increases the number of peripheral epithelial buds [30] .', 'Addition of exogenous BMP4 protein to mesenchyme-free endoderm cultured in vitro with FGFs , on the other hand , inhibits proliferation , secondary budding and differentiation [28] , [29] , [31] .', 'Both overexpression of BMP4 and a conditional knock-out result in similar lung phenotypes [32] , [33] , suggesting that correct BMP4 levels are essential for normal lung development .', 'These may be maintained by the many negative feedbacks that control BMP activity .', 'Theoretical studies suggest that physical forces can play a key role in lung branching morphogenesis [34]–[37] .', 'Recent studies demonstrate that the mechanical stresses do influence branching morphogenesis [38] .', 'Increased internal pressure leads to an increase of lung branching in in vitro cultures [39] , and cellular contractility is critical for branching morphogenesis of the lung .', 'Inhibiting actomyosin-mediated contractility in whole lung explants decreases branching [40] , whereas activating contractility increases branching [41] .', 'A qualitative model that described epithelial branching in culture experiments showed that both the mechanical strength of the cytoskeleton and the reaction-diffusion kinetics can in principle affect branching morphogenesis [36] , [42] .', 'Work on mammary glands further suggests that the geometry of tubules might dictate the position of branches [43] .', 'Computational models can explore the impact of the signaling interactions , physical forces and domain geometries and thus discern a minimal set of rules and interactions from which the observed pattern can emerge .', 'Hirashima and co-workers recently proposed a simple three component model on a 2-dimensional lung bud cross-section [44] to explain the mechanistic basis of different branching modes .', 'The model focused on the interactions between SHH , transforming growth factor ( TGF ) -beta and FGF10 and suggests that domain length and shape can have a strong impact on the distribution of morphogenes and the selection of branching points in the developing lung .', 'This prediction , however , hinges on a particular distribution of TGF-beta ( constant at the stalk ) and SHH ( fixed at the tip ) and is valid only with a particular type of boundary condition ( impermeable lung boundaries ) which is unlikely to apply ( Figure S1 ) .', 'Turing-type models have been suggested as an alternative to explain the emergence of regular patterns as observed during lung branching morphogenesis [45] , [46] .', 'However , to our knowledge no such mechanistic Turing model has yet been formulated for the lung .', 'Here we present a reaction-diffusion model that we developed based on available information from the literature .', 'The Turing-type model reproduces available experimental data both in wild type and mutant mice and provides a mechanistic explanation for the different lung branching types .', 'We further show that the rate with which the lung bud grows can determine the branching mode .'], ['The lung tip can be represented as a cylinder with an internal radius of and an external radius of 100 ; the thickness of the lung epithelium can be estimated from microscopy data as 5–10 [32] , [59] .', 'To keep the calculations as simple as possible while retaining a realistic geometry we use a 2D slice in Cartesian coordinates along the lung bud axis of symmetry as shown in Figure 1b .', 'SHH is produced in the epithelium [1] , [2] and FGF and Ptc are produced in the surrounding mesenchyme [1] , [2] .', 'There is no experimental evidence that the mesenchyme or epithelium are surrounded by any insulating layer and accordingly the boundaries are permeable in our simulations .', 'SHH and FGF can thus diffuse unhindered outside of the epithelium and the mesenchyme as well as in the lumen which is filled with a liquid .', 'The unhindered diffusion of SHH and FGF in the lumen and interstitial space is reflected by the diffusion coefficient which is much larger than the diffusion coefficients , , that apply in the tissue; the receptor Ptc is a membrane protein and therefore cannot diffuse into the cavities .', 'The concentrations of FGF and SHH infinitely far from the mesenchyme are assumed to be zero .', 'We note that the predictions of our model are independent of the boundary conditions and similar results are obtained in simulations with zero-flux boundary conditions at the boundary of the lung tissue ( see section Robustness of the Observed Pattern for details ) .', 'We start all our simulations with no species present , i . e . we are setting all concentrations to zero at .', 'We consider two modes of 1-dimensional growth along the lung stalk: ( 1 ) uniform growth ( stretching ) of the domain and ( 2 ) local growth at the tip of the lung .', 'To conserve mass ( rather than concentration ) the reaction-diffusion equations ( Eqs 6–8 ) must be expanded to include the advection and dilution terms [60] , i . e . ( 9 ) where denotes the growth speed .', 'For homogeneous growth at rate we then have ( 10 ) where denotes the initial height of the lung tip , the height of the lung tip , and denotes time .', 'The domain size and the time scale of the process are well established [17] , [24] .', 'The measurement of the in vivo kinetic parameter values on the other hand is complicated and typically parameter values are known only from experiments in related systems [52]–[54] .', 'To reduce the number of unknowns we non-dimensionalize the equations and thereby remove five independent parameters .', 'We use , the internal radius of the lung bud , as characteristic length scale of the model and as its characteristic time scale .', 'Moreover , we non-dimensionalize the FGF concentration with respect to the Hill constant , i . e . , and the SHH and Ptc-1 concentrations with respect to , i . e . , with .', 'Equations 6–8 can then be rewritten in dimensionless form; the dimensionless parameters and variables are summarised in Table S1 . ( 11 ) It should be noted that the Laplacian is now with respect to the non-dimensional space variables and .', 'Five parameters have been removed and the patterning mechanism no longer depends on absolute values of diffusion and decay constants , but only on the relative diffusion coefficients and the relative decay rates .', 'Similarly , the absolute concentrations do not matter , but only the relative concentrations ( as a result from the relative expression and decay rates ) relative to the Hill constants and the effective binding constant .', 'The value of the Hill coefficient was set to to account for possible cooperatively effects; however , the model gives very similar results with other values of ( Figure S4 ) .', 'It should be noted here that the stochiometry of SHH and Ptc complex at which patterning is observed is not limited to the case m\u200a=\u200a2 , n\u200a=\u200a1 that we analyse in this manuscript .', 'Similar patterns are observed as long as , ( see Figure S5 for details ) .', 'The PDEs were solved with finite element methods as implemented in COMSOL Multiphysics 4 . 1 .', 'COMSOL Multiphysics is a well-established software package and several studies confirm that COMSOL provides accurate solutions to reaction-diffusion equations both on constant [61] and growing two-dimensional domains [62]–[64] .', 'Mesh and the time step were refined until further refinement no longer resulted in noticeable improvements as judged by the eye ( Figure S6 ) .', 'When simulations were performed on an open domain the bulk solution conditions was implemented at a distance from the mesenchyme , where is the maximum time of model evaluation .', 'It was shown that beyond this , the effects of diffusion in not important on the experimental time scale [65] .', 'A local stability analysis was performed in the following way: parameter values were taken as indicated in Table 1 , further parameters were varied one by one until qualitative change of pattern were observed ( different number of FGF , SHH and Ptc spots ) .', 'Accuracy of parameter value estimation was 10% or higher .', 'The approach to estimate robustness to spatial parameter variability was adapted from ref [66] .', 'Parameter values were assumed to be given by the formula , where is normally distributed random function with a mean value of zero and half width .', 'The half width of the distribution was equal for all parameters , except geometrical which were not varied .'], ['Since the model is an example of a Schnakenberg Turing-type model [58] we expected to find parameter ranges for which we would observe the emergence of FGF patterns on the lung-shaped domain .', 'Indeed such pattern emerged from spatially uniform initial conditions .', 'We note that the observed distribution and expression pattern correspond overall rather well to experimental observations , but since quantitative data on protein concentration distributions are not available we have to restrict ourselves to a qualitative discussion of distribution patterns .', 'We therefore deliberately left out scale bars for better readability .', 'Dark red colours denote the highest concentrations while dark blue colours denote the lowest concentrations .', 'FGF10 induces SHH expression and SHH expression is highest in the epithelium adjacent to the mesenchyme with high FGF10 concentrations ( Figure S7b , e ) as also observed in experiments [24] , [32] , [48] , [67] .', 'In situ data seem to suggest that the expression domains of Ptc-1 and Shh coincide [24] , [68] , [69] but the spatial resolution of the data may be too low to reveal the closed juxtaposition predicted by the model .', 'Different parameter ranges resulted in steady-state patterns with FGF10 either centered at the lung bud tip ( Figure 2a ) or concentrated towards the side of the tip ( Figure 2d ) .', 'Experiments have previously demonstrated that lung tips grow towards regions with a high FGF10 concentration [59] and accordingly FGF10 centered at the lung tip will support further outgrowth of the lung tip while FGF10 centered at the sides will induce lateral outgrowth .', 'We thus suggest that the two types of patterns that we observe are likely to correspond to the different branching modes , i . e . lateral branching would thus correspond to the FGF10 pattern in Figure 2a with FGF10 spots at the tip and at the side , while bifurcation would correspond to the FGF10 pattern in Figure 2d with FGF10 centered only to the side but towards the tip .', 'Our model cannot reproduce domain branching or the difference between planar and orthogonal bifurcations as these events are intrinsically three dimensional .', 'Here we should note that the model generates pattern similar to those discussed above if extended to the third dimension ( Figure S8 ) .', 'However , a much wider range of patterns is possible in 3D as will be discussed in detail in a separate manuscript ( Menshykau and Iber , in preparation ) .', 'Going forward we will restrict the presentation to the FGF10 concentration pattern as these appear to mainly guide lung outgrowth .', 'Since the parameters are difficult to determine accurately in experiments it was important to check how sensitive our results would be to variations in parameter values .', 'We carried out a local stability analysis by altering each parameter value independently .', 'Here we note that the non-dimensional parameters represent relative dimensional parameters .', 'The patterning mechanism is thus robust to a parallel change in the parameter pairs listed in Table S1 , and the value of most relative parameters can be changed by 20%–30% without changing the type of the observed pattern ( Figure 3 ) .', 'At the same time almost each parameter ( except for the degradation rate of the receptor , ) can be employed to switch the pattern between lateral branching ( Figure 3 , blue ) and bifurcation mode ( Figure 3 , red ) .', 'The mechanism thus appears to be robust to noise yet sensitive to regulation .', 'We note that virtually all parameter values could be affected by the many additional interactions that have been described but that we chose to ignore in this simple model that focuses only on the core of the regulatory mechanism .', 'The domain that we chose to solve our model on is an idealization of realistic domains .', 'We therefore also checked the impact of domain deformations ( Figure 4 ) .', 'The lateral branching mode is indeed robust to deformations of the domain geometry , i . e . to an increased radius of the epithelium ( Figure 4a ) , the mesenchyme ( Figure 4b ) or a truncated bud ( Figure 4c ) .', 'Similarly , the boundary conditions are not critical and similar pattern emerge with zero-flux boundary conditions both in the lateral ( Figure 4', 'd ) and in the bifurcation mode ( Figure 4', 'e ) .', 'Linear stability analysis of Equation 11 carried out with parameter values as used to simulate lung branching on the domain with no-flux boundary conditions ( Figure 3 d ,', 'e ) showed that pattern arise because of the diffusion driven instability and therefore Turing type .', 'Since lung buds branch as they are growing out we wondered how the growth speed and type would affect patterning .', 'Figure 5 shows the FGF10 distribution on a growing lung bud where growth is restricted to the lung tip .', 'At the concentration of FGF is high at the tip of the growing lung , and further regions of high concentration of FGF10 appear at the lung stalk as the lung grows out ( Figure 5', 'a ) .', 'Importantly new regions of high FGF10 concentration emerge close to the tip .', 'This pattern corresponds to the lateral branching mode .', 'At a 4-fold lower growth speed , ( Figure 5', 'b ) , regions of high FGF concentration appear only at the sides of the lung tip but are absent from the tip itself .', 'This pattern would thus correspond to the bifurcation mode of branching .', 'If the lung bud grows uniformly in the entire domain the patterning is similar with an important difference: regions with high FGF10 concentration can appear at any position within the domain ( Figure S9 ) .', 'Both the insertion of branches directly behind the tip and the insertion of new branches at the proximal side of the stalk have been observed .', 'Similarly , both uniform proliferation ( Metzger and co-workers , personal communication ) , [17] and enhanced proliferation at the tip [70] have been observed .', 'Those experiments that report a concentration of proliferating cells at the tip find 2 . 5 times more BrdU-stained cells ( a marker for dividing cells ) at the tip than in the stalk [70] , and in our model branching points appear behind the tip as long as the growth rate in the tip is at least 2–3 times larger than in the stalk .', 'We therefore suggest that different lung branches may be growing differently , and that this may explain the different positions at which new branches emerge relative to the tip .', 'Earlier we showed that the pattern on a constant domain is robust to small parameter variations .', 'To test that the pattern is robust to small random changes also on growing domains we explored the patterning mechanism if parameter values are drawn from a Gaussian distribution with different standard deviations ( see Model section ) .', 'Figure S10 shows that the domain branching mode remains stable as long as standard deviations of the random variables do not exceed 0 . 2–0 . 3 of the reference value presented in Table 1 .', 'An important test for the suitability of a mathematical model is its consistency with a wide range of independent experimental observations .', 'Lung branching morphogenesis has been studied intensively and a large body of experimental results exists to test the model with .', 'These include a large number of in part counterintuitive mutant phenotypes of key signaling proteins in mice [1] .', 'Since our model is restricted to FGF , SHH and Ptc-1 we will focus on mutations in those genes .', 'The computational model reproduces all full knock-outs .', 'However , these results are trivial since the lungs in mutants with fully knocked-out genes are severely truncated or do not form at all [7] , [49] , [71]–[73] .', 'Given the key importance of all three components in the model no pattern is observed if any component is eliminated .'], ['We have developed a model of the core signaling interactions that governs lung branching morphogenesis ( Figure 1a ) .', 'We find that the experimentally described interactions give rise to a Schnakenberg-type Turing patterning mechanism which leads to the self-organized emergence of FGF10 pattern from homogenous initial conditions during in silico lung bud development .', 'The predicted pattern are overall consistent with experimentally observed expression pattern both in wildtype and mutants , and the model predicts the reported increased spacing between FGF10 pattern in mutants with reduced FGF10 expression .', 'The latter is a non-trivial prediction that cannot be explained with models reported earlier [36] , [42]–[44] .', 'The parameter values are largely unknown for the developing lung , but many of the parameters have been established in other model systems ( Table 2 ) .', 'To reduce the number of unknown parameters we non-dimensionalized the model and our conclusions thus only hinge on relative parameter values .', 'In particular , we require that SHH and FGF10 diffuse about a 100-times faster than the receptor Ptc .', 'Receptors are membrane proteins and are known to diffuse some 100–1000-fold more slowly than proteins in solution [55] , [56] .', 'The diffusion of SHH and FGF10 in the fluid-filled cavities , , is again some 10-fold higher as is common for unhindered diffusion of proteins [81] .', 'Since the domain size and the time span of the developmental processes are well known also the absolute values of the parameters can be estimated from their non-dimensional counterparts ( Table 2 ) , and we note that all parameter values are within physiological ranges established in other model organisms , i . e . [52] , [53] , [81] .', 'Finally we note that introducing the simplifying quasi-steadystate assumption for the formation of the SHH-Ptc complex does not affect the observed pattern as long as the non-dimensional binding and unbinding constants are of order 1 which is in the likely physiological range .', 'The model shows that the different branching modes could result from different patterns of FGF10 .', 'Thus for certain parameter ranges FGF10 accumulates at the tip of the lung bud .', 'Since FGF10 induces outgrowth towards the highest FGF10 concentration this should lead to an elongation of the lung bud .', 'As the lung bud is elongating at the tip new FGF10 spots emerge at the stalk of the lung bud close to the tip .', 'These spots would lead to the lateral outgrowth of the lung bud and thus to lateral branching .', 'For other parameter ranges FGF10 is absent from the lung tip and concentrates towards the sides .', 'The lung bud would thus no longer elongate but grow out towards the sides which could be interpreted as the bifurcation branching mode .', 'Since we only consider a 2D slice of the lung bud the model cannot differentiate between planar and orthogonal bifurcations .', 'Interestingly , most parameter values can alter the FGF10 distribution and thus the mode of branching ( Figure 3 ) .', 'An increase in the diffusion constants or a lowering in domain size favour bifurcations while enhanced protein production or reduced decay tend to favour lateral branching .', 'This may explain why genetically different individuals tend to have different branching patterns .', 'At the same time this offers regulatory control to the many signaling factors that can affect these rates , but which we chose to ignore in this simple model that focuses on the core signaling proteins .', 'Thus BMP signaling may lower the rate of FGF10 expression or interfere with the FGF-dependent increase in SHH expression .', 'Both effects would favour bifurcations over lateral branching .', 'BMP-dependent induction of Gremlin and Noggin on the other hand would lead to the sequestration of BMPs and thus limit the impact of BMPs .', 'A strong increase or decrease in BMP activity may then move the SHH and FGF10 expression rates outside the patterning range .', 'This may explain the detrimental impact of both Bmp overexpression and conditional knock-out on lung branching morphogenesis [32] , [33] .', 'Further computational modelling combined with experimentation will be required to clarify this .', 'The effective diffusion constants may be affected by the expression of glycoproteins as previously reported for the morphogen Dpp [82] .', 'Growth has previously been reported to strongly affect pattern selection in Turing models [83] .', 'Indeed the rate of growth also alters the branching pattern in our simulations ( Figure 5 ) .', 'Thus we observe FGF10 pattern characteristic of lateral branching at high growth speeds and FGF10 pattern characteristic of bifurcations at low growth speeds .', 'The growth rate used to model the lateral branching mode in our model is around 14 and gives rise to two new branches per day with branches separated by approximately 150–200 ; this is well in agreement with experimental observations [17] , [24] .', 'To simulate the bifurcation mode of branching we use a growth rate of 3 . 6 which is close to the growth speed estimated from data in Metzger and co-workers [17] .', 'We can speculate that the concentration of FGF10 affects the speed of outgrowth such that changes in the concentration during outgrowth would determine the sequence of the branching events .', 'For our particular choice of parameters we observe lower FGF10 concentrations in spots close to the tips of rapidly growing lung buds ( Figure 7 ) which may lead to bifurcations in the next generation .', 'More proximal lateral spots have higher FGF10 concentrations , and these branches may grow out to longer length as indeed observed for some of the proximal branches .', 'We further note that the position at which new branches appear depends on the growth mode .', 'Thus if growth is at least 2–3 times faster at the tip then new FGF10 spots emerge directly behind the tip .', 'In case of more uniform growth FGF10 spots appear also in the more proximal domain .', 'Both growth modes and patterning dynamics have been observed during lung branching morphogenesis , and we therefore suggest that both growth modes exist in the developing lung and lead to the different patterning sequences .', 'Further advancements in our understanding of lung branching morphogenesis will require the development of three-dimensional models where the local growth rate is coupled to the FGF10 concentration and the inclusion of more signaling factors , most importantly those of BMPs .', 'The parameterization and validation of such models will require new experimental data that reveal the dynamics of the three dimensional dynamics of branching and that quantify the epithelial and mesenchymal responses as well as the growth speed relative to the FGF10 concentration .', 'Such information can now be acquired with the help of optical projection tomographs [84] , and this method has been already used to capture the three dimensional dynamics of developing kidney [85] .', 'Further advances in experimental techniques can thus be expected to provide exciting new insights into the regulatory processes of branching morphogenesis during organ development .']]
['Introduction', 'Model', 'Results', 'Discussion']
['Many organs of higher organisms , such as the vascular system , lung , kidney , pancreas , liver and glands , are heavily branched structures .', 'The branching process during lung development has been studied in great detail and is remarkably stereotyped .', 'The branched tree is generated by the sequential , non-random use of three geometrically simple modes of branching ( domain branching , planar and orthogonal bifurcation ) .', 'While many regulatory components and local interactions have been defined an integrated understanding of the regulatory network that controls the branching process is lacking .', 'We have developed a deterministic , spatio-temporal differential-equation based model of the core signaling network that governs lung branching morphogenesis .', 'The model focuses on the two key signaling factors that have been identified in experiments , fibroblast growth factor ( FGF10 ) and sonic hedgehog ( SHH ) as well as the SHH receptor patched ( Ptc ) .', 'We show that the reported biochemical interactions give rise to a Schnakenberg-type Turing patterning mechanisms that allows us to reproduce experimental observations in wildtype and mutant mice .', 'The kinetic parameters as well as the domain shape are based on experimental data where available .', 'The developed model is robust to small absolute and large relative changes in the parameter values .', 'At the same time there is a strong regulatory potential in that the switching between branching modes can be achieved by targeted changes in the parameter values .', 'We note that the sequence of different branching events may also be the result of different growth speeds: fast growth triggers lateral branching while slow growth favours bifurcations in our model .', 'We conclude that the FGF10-SHH-Ptc1 module is sufficient to generate pattern that correspond to the observed branching modes .']
['Most organs of higher organisms , such as the vascular system , lung , kidney , pancreas , liver and glands , are heavily branched structures .', 'The branching process during lung development has been studied in great detail and is remarkably stereotyped .', 'The branched tree is generated by the sequential , non-random use of three geometrically simple modes of branching .', 'While the branching sequence is identical in mice of identical genetic background it differs between mouse strains .', 'This suggests that the positioning of branch points and the type of branching sensitively depends on information encoded in the genome .', 'Encoding every branching point independently in the genome would require a large number of genes , and it is more likely that a recursive , self-organized process exists that determines the patterning .', 'While many regulatory molecules have been identified an integrated understanding of the regulatory network ( program ) is missing .', 'Based on available experimental data we have developed a model for lung branching .', 'The model correctly predicts branching phenotypes in mutants and suggests that also the growth speed of the lung tip can affect the positioning and type of the next branching event .']
['developmental biology', 'organism development', 'biology', 'computational biology']