Abstract. Introduction

Transcription

1 A coupled morphogen kinetics and viscoelastic tissue mechanics model of the developing chick limb Ben Jordan +, Eric Richardson, Suzanne Haydon Department(s) of Mathematics +, Biomedical Engineering, and Biochemistry, Molecular Biology, and Biophysics University of Minnesota, Minneapolis, MN May 5, 2007, v0.3 Abstract The chick forelimb is a model system in the study of the biological processes involved in tissue growth and pattern formation [50]. The manipulation of these mechanisms in numerous experimental studies [6, 38, 40, 41] has resulted in the elucidation of many qualitative, as well as quantitative aspects involved. We use this information to formulate a temporospatial model for growth based on the coupling of the intracellular and extracellular kinetics with the tissue mechanics. The resulting systems of non-linear partial differential equations are solved iteratively over time within an approximation of the geometry of the stage 16 (HH16) limb bud. The resulting code, data, and visualizations provide a flexible and unique method for the investigation of further experimental questions in limb development. Introduction Limb development in the chicken is produced by a complex system of morphogens and gene products that regulate the growth and determination of cells in the various regions of the limb[55]. Three regions in the growing limb are of special importance: the zone of polarizing activity (ZPA), which determines anterior-posterior patterning; the apical ectodermal ridge (AER), which determines proximal-distal growth[5]; and the dorsal ectoderm, which produces, amongst other things, the Wnt family of morphogens, which help determine dorsal-ventral patterns (see figure 1) [55],[19]. These regions serve as organizing centers where several morphogens are produced and then diffuse into the surrounding tissue where they activate or inhibit a number of gene products[50]. These gene products in turn regulate other gene products and ultimately determine the fate of each cell. A vast number of experimental studies have been done in the chick limb[54] and other model systems to determine the regulation pathways of these morphogens. Several types of experiments have been used to study these pathways, including gene knockout studies in mice, transplantation or substitution of the AER and ZPA in chick[55], implantation of beads soaked in a specific 1

2 morphogen[75], and labeling expression domains through in situ hybridization[37]. These experimental studies continue to find new genes and gene products involved in the patterning of the limb. Sonic hedgehog (Shh) is one of the first and most studied morphogens in this system. Experiments have shown that Shh originates in the ZPA and contributes to the pattern of digits in a concentration and time dependent manner[53],[45]. Hox genes, dhand, and mario are a few of the genes known to be involved in specifying digit pattern in an Shh dependent manner[18],[2],[29]. Defects in Shh signaling lead to the formation of extra digits even though Shh levels are normal[50], suggesting that intermediaries, like Shh receptors, are also important factors in this pathway. Another set of important morphogens for limb development are the fibroblast growth factors (FGFs), which originate in the AER and are crucial for proximal-distal (shoulder to tip) growth. At least one of these FGFs, FGF4, is part of a positive feedback loop with SHH, using Gremlin (GRE) as an intermediate[33],[54]. FGF8 also affects SHH levels, though it does not form a feedback loop with it[42]. While at least six FGFs are involved in limb growth, only FGF10, FGF8 and FGF4 have been well studied. Wnt signaling is another key pathway in limb formation. At least two different Wnts are involved in patterning the wing starting with bud initiation. Wnt7a works in Dorsal-Ventral patterning and regulation of Shh, while Wnt3a is needed to maintain the AER[19]. The Wnt signaling pathway involves a number of species, such as beta-catenin and Frizzled receptors, that are not considered in the current limb model. The mechanics of a growing limb presents an equally complex system. Stress and pressure distributions within a growing tissue may determine the growth and death of cells, such as in tumors. Stress distributions in tumors determine predictable layers of viable and necrotic tissues. While a cell may receive the proper chemical signals to divide or grow, this could be inhibited by local stresses. In addition, cells near a curved surface may exert an effective surface tension on cells in the interior, stunting growth. Modeling the deformations and stresses of a material requires the use of continuum mechanics. Continuum mechanics makes the assumption that a material can be represented by infinitesimally small elements with certain mechanical properties. Most metals, for example, can be modeled by assuming that each of these elements acts as a spring. In a spring, the resistive force given by the spring is directly proportion to the elongation or compression that it undergoes (Hooke s Law). Spring-like mechanical properties are also known as linear elastic properties. With linear elastic properties, no time dependence is involved; the instantaneous elongation/compression is directly correlated with instantaneous force regardless of the spring s history. Most biological tissues, however, are modeled not as a linear elastic materials, but as viscoelastic materials. Each element in a viscoelastic material is described by a combination of both springs and dashpots, as shown in figure 2. Dashpots add interesting characteristics to the model. The resistive force given by the dashpot is directly proportional to the rate of elongation or compression. This 2

3 adds dependency on time. Therefore, the force that a viscoelastic unit exerts on its neighbors depends on its history of compression or elongation. For example, if a viscoelastic material is suddenly compressed, it will exert very large forces on its neighbors. The dashpot(s) will then relax, and the remaining force will be due to the springs in the model. Two terms are now introduced for the sake of clarity: stress is defined as a force divided by the area over which it acts. Strain is the elongation or compression of an element divided by its original length. Assuming that the area and original length do not change, stress is directly proportional to force, and strain is directly proportional to elongation. Therefore, if force and elongation are directly proportional, so are the stress and strain. Both sets of terms are used interchangeably in this paper. After an appropriate model is selected, a constitutive equation can be derived that describes the relationship between stress and strain over time in an element. The equation can be then extrapolated to the whole material (or continuum) using the principles of continuum mechanics. This usually results in a differential equation that describes the mechanics of the material with time. Specific Aims Simulation of knockout model experiments with the Reaction-Diffusion model. As stated in the introduction, various knockout models have provided information about the roles of morphogens in limb development[50]. We proposed to simulate these experiments in the R-D model by inhibiting or removing certain morphogens in the model, and then observing the outcome. Through this we hoped to validate the accuracy and stability our R-D model before integrating it with the mechanics. Development of an accurate 3D mesh to approximate the budding limb. Before the mechanics aspect was solved, a discrete mesh with appropriate dimensions was needed. Important requirements for this mesh were to create defined areas within the mesh to introduce morphogens (AER, ZPA, etc), refining the mesh in areas that would see large amounts of growth, and properly constraining the mesh with boundary conditions (both mechanical and chemical). Integration of the Reaction-Diffusion model with the Finite Element growth model to produce a functional model of limb growth. After we successfully created both models, we proposed to integrate them into a fully functional model. We needed to decide what combination of morphogens (and for what duration of time) determines an appropriate growth signal to the mechanics model. The mechanics model should respond appropriately by growing in those areas where is receives such the growth signal. 3

4 Methods Reaction-Diffusion Kinetics Model We started by creating a list of relevant papers which give experimental results for interactions involving the 16 morphogens in our model. These papers included both knockout and bead implantation experiments done in both the chick and mouse limb, although preference was given to data from chick limbs as there are some known, and possibly a lot of unknown, differences in the two systems. From these we were able to compile a list of parameters for comparison to our model (see SOM). To test the accuracy and stability of the R-D Model we made changes to individual equations (see below for equations) in our code and ran them in Matlab. The equation for the morphogen being knocked out was set to zero. This produced a flat line at zero in our Matlab graph for that morphogen. We could then look at changes in the graphs of the other morphogens in the network. Most experiments were run for 10 hours with a time increment of 0.1 hours and a few were rerun for only one hour, with a time increment of 0.01 hours. Morphogen Production Equations F FIM = k F IM (1) F ZP A = k S (F 4 + α 1 F 8 )W K S + (F 4 + α 1 F 8 )W F F4 = k F K F 4 Bmp (2) (3) F F8 = k F 8 F 10 K F 8 + F 10 (4) F W nt = k W nt (5) 4

5 Reaction-Diffusion Equations (Dimensional) F IM t F 10 t = δ FIM 2 F IM k FIM F IM + F FIM 1 θ f θ f χ IM (6) = δ F10 2 F θ f θ f k F 10 F IM K F 10 + F IM k F10 F 10 (7) F 8 t = δ F8 2 F 8 k F8 F 8 + F F8 1 θ f θ f χ AER (8) S t = δ S 2 S + γ( k + sps P + k spsp ) k S S + F Z (9) W t = δ W 2 (W ) k W W + F W nt 1 θ f θ f χ ectod (10) P t = k+ sps P +k spsp k + mpm a P +(k mp+k mp )MP +k 0 p+k pm a k P P (11) SP t = k + sps P (k sp + k sp )SP k + spmm i SP + (k spm + k spm )SP M (12) Gre t Bmp t M a t M i t = k + mpm a P + k mpmp + k spm SP M (13) = k + spmm i SP + k spmsp M + k mp MP (14) MP t SP M t F 4 t G3a t = k + mpm a P (k mp + k mp )MP (15) = k + spmm i SP (k spm + k spm )SP M (16) = δ F4 2 F 4 k F4 F 4 + F F4 1 θ f θ f χ AER (17) G3r t = k G3r G3a 1 + K g3r M a k G3rG3r (18) = k g3a k G3r G3a 1 + K g3r M a k G3aG3a (19) = δ Gre 2 Gre + 1 θ f θ f = δ Bmp 2 Bmp + 1 θ f θ f k gre 1 + K greg3r k GreGre (20) k Bmp 1 + K Bmp Gre k BmpBmp (21) 5

6 Initial tests were made to determine if these knockouts were affecting the correct downstream species based on the morphogen network (figure 3) created for the set of equations used. This was done by starting with the most upstream morphogen (Fgf10) and progressively moving downstream until all species were knocked out. If any knockouts did not correctly affect morphogens directly downstream, the code was checked for errors. Next, the Matlab graphs of the knockout experiments were compared to results complied from experimental papers to determine the accuracy of our model. Since performing all possible combinations of knockout experiments was impractical, we focused on a few that we thought would produce interesting results and ones that we could compare to the available literature. A few bead implantation experiments were also simulated by changing the initial concentration of morphogen to that used in the bead. Since the bead implantations are often done in regions of the chick limb not endogenously expressing the morphogen in the bead, we had to remove the production term from the equation and turn off any upstream species that are also absent in that region. The values for the rate constants and other parameters are not included in this report (see SOM) Assumptions Since the R-D equations were taken from a previously existing code, a number of our assumptions are described there. The most important ones in our testing of the stability of the model are: 1) No diffusion is taking place as the local dynamics R-D Model we started with is in a single cell. Diffusion terms in the equations were removed for looking at local dynamics. Diffusion terms will be used in the 3D Model. 2) Transcription and translation rates of genes and gene products are not separately taken into account, but are combined into the production rate. 3) No post-translational modifications to the morphogens are accounted for. This is specifically applicable to Shh which undergoes both a cleavage event and a cholesterol modification before diffusing in vivo[12],[47]. 4) We assume that knockout experiments in mice can give us information about ways the network works in chicks. Parameters Used Parameters of production, initial concentration, and binding rates had previously been determined in the R-D Model. We initially assumed these to be correct parameters for testing the stability of the model. As we read more chick limb experimental papers, we checked the accuracy of these presumed parameters and made changes where necessary. 6

7 Viscoelastic Tissue Mechanics As for the mechanics component of our model, Mech3D uses a continuum mechanics/fe approach to describe solve for strains and stresses. It does this through calculating the deformation gradient, F. The deformation gradient of each time step is decomposed into two parts, F G (deformation due to tissue growth), and F V (deformation due to mechanics stresses). Briefly, after the reaction-diffusion equations are solved for concentrations of morphogens at every point within the tissue, S, which is the volumetric source term, is determined at each node. S can be a function of not only of morphogen concentration, but stress at that point as calculated in the previous time step. The S term is then added to the model through the F G component. If growth is uniform, there are no internal stresses in the material, and only the F G component is needed. However, because growth is often not uniform, F V (the mechanics component) must resolve discontinuities in stresses and pressures (see Figure 4). The constitutive equation, which describes how each element reacts to mechanical forces, is based on a cross between the Kelvin-Voight and Maxwell models. The equation is presented in figure 5. Through various manipulations, we generalize this one-dimensional model to three dimensions using continuum mechanics. The final constitutive equation is also shown in figure 5. There are several assumptions in the above equations, such as incompressibility, lack of thermal effects, and negligible acceleration of the tissue. A complete description of assumptions, as well as proper mass, momentum, and thermodynamic equilibrium equations were described by Stolarska, et. al [66, 61]. The constituitive equation, the RD equations, mass balance equation, linear momentum balance equation (eq.4), angular momentum balance equation and energy balance equations (Figure 5) are solved by the Mech3D code at each node during each time step. This produces the pressures, stresses, displacements and morphogen concentrations at finite points within the growing mesh. It should be noted that one of our aims in our original proposal stated that we would generate a 2D mechanics model as a preliminary step. We quickly learned that the task of modifying the complex code would not be feasible, so we decided to work with the existing code in 3D. It should be noted that while Mech3D supplied the stress tensor at each point (3 normal stresses, 3 shear stresses), an average stress was needed during analysis of the data. The Von Mises stress, commonly used in facture analysis to find the largest stress in a material, was calculated to fulfill this need. The Von Mises stress takes each stress tensor component into account, and is calculated using the equation 7 shown in figure 5. Meshing and Finite Element Method The approximation of the solution of this system is implemented with the finite element (FE) method. This technique allows for the system to be solved inside of a complex, dynamic geometry, such as the growing limb bud. The 7

8 discretization of the volumes is handled by a Delaunay meshing algorithm, restricted to the use of tetrahedral and triangular element types. The various tissues, signaling centers, and bead implantation sites are created as volumes with unique indexes. These morphological features are adjustable within the parameters of the code to allow for variation in organismal phenotype or ectopic expression of chemical species. Resulting is a set of nodes, elements, and volumes with which we can approximate the changes in chemical concentrations, stresses, strains, pressures, and displacements in our tissue geometry as it changes over time. The numerical solution to the both the reaction-diffusion kinetics and the viscoelastic mechanics problem is handled by a non-linear PDE solver which employs an iterative Newton s method. The values of the previously listed quantities at each element, as well as it s nodal coordinates are output at each time step. Computational Design and Implementation Mech3D, a code for the solving of coupled RD and viscoelastic tissue mechanics, was modified to model limb development. Prior applications have been on tumor modeling and nutrient transport. The code is complex, with several subroutines and many references to finite element and meshing libraries. A complete guide to the code is found in the appendix, along with a copy of the code. Briefly, the driver program (mech3d.f90) calls in both general problem data (parameters global.limb.in) and boundary data (parameters bdry.limb.in). A mesh, previously created by GMESH as explained earlier, is also read into the program. Memory is then allocated, boundary and initial conditions are set, and Fdriver.f90 is called to coordinate the solving of both RD and mechanics equations. First, the RD equations contained in Flimb3d react.f90 are solved with NITSOL (a nonlinear solver) using the corresponding parameters and conditions in parameters react.limb.in. The mechanics equations in Flimb3d mech.f90 are then solved using NITSOL and the parameters and conditions in parameters mech.f90. In addition, the growth component based on the results of the RD equations is solved and incorporated into the deformation gradient. Finally, the solution is written to an output file. A diagram describing the program hierarchy is found below. 8

9 Results RD Knockout Experiments We were able to simulate knockouts of each gene product in the R-D model by setting equations equal to zero, which gives a flat line at zero on our Matlab graphs. Knockouts were simulated in this way because in vivo they are done by inactivating the gene for a specific morphogen, usually by mutation, so there will be no initial concentration or production of that morphogen. With all the species active, most reached steady state within ten hours (Fig. 6) and all interesting fluctuations in knockout experiments also happened within this time frame, so most experiments were only run for 10 hours to make changes easier to see. A few experiments that seemed to have interesting fluctuations within the first hour were rerun for a closer look at those fluctuations (Fig. 7). All changes in morphogen levels were determined by comparison to Fig. 6. We started with testing knockouts of Fgfs (Fgf10, Fgf8, and Fgf4) and Wnt as these species are the farthest upstream in our network. For example, an Fgf10 knockout greatly reduced Fgf8 levels from wild type (Fig. 8), as it should based on the morphogen network. While these knockouts affected the species directly downstream as expected, only knockouts that changed the Shh level affected species downstream of Shh as the entire network gets funneled through Shh (Fig. 3 and Fig. 9). The level of Shh undergoes the greatest change when three of the four (Fgf10, Fgf8, Fgf4, Wnt) species that feed directly into Shh are knocked out, although a Wnt knockout alone will affect Shh enough to also make downstream changes. On the network map these four feed into the ZPA, where Shh is produced, and changes to Shh can also be made by knocking out the ZPA (Fig. 10). As progressive knockouts through the network were made, correct downstream effects were looked for. The working system was best seen through a series of knockouts from Smoothened active (Ma) through Gli3R, Gremlin, and BMP to Fgf4 (fig. 11). Since each of these species inhibits the one directly downstream of it, the knockouts give alternating rises and falls in each downstream species. Gli3-a was the least affected by changes in the other species in the system. A number of the species were hard to test for accuracy as they have more complex relationships with the other species in the network (for example, Smoothened active which interacts with five other species). One problem with the equations is evident here, when Shh or the ZPA are knocked out, there should be no Shh present in the system. However, if any of the complexes containing Shh have an initial value set greater than zero and the equation for Shh is not set to zero, then there will be a small spike in Shh early on (Fig. 12). Either the model will need to be corrected for this, or future experiments will need to have any complexes manually turned off during knockouts. Manual removal of complexes was done for the comparisons to in vivo experiments. We then compared out knockout experiments with those done in chick or mouse limbs. When Shh is knocked out in mice, Fgf4 expression is not main- 9

10 tained but Fgf8 expression is normal[42]. A knockout of Shh in our R-D model does leave Fgf8 at a normal level, but Fgf4 is only lessened slightly (fig. 13). When Fgf4 is removed in mice, levels of Shh, BMP, Fgf8 and Fgf10 are unchanged and the limb grows normally[27],[42]. We can also achieve this result in our R-D model (fig. 14). Shh and Gre are not expressed in the same cells in the chick limb and here the signaling network relies on the distance Shh can diffuse. At a certain stage of growth, these two cell populations get far enough apart to stop the feedback loop[33]. While we couldn t simulate the limit of diffusion in the local dynamics R-D Model, we could simulate this by knocking out Shh and seeing levels of both Gre and Fgf4 decrease slightly (fig. 13). A Gli3 knockout in mice leads to polydactyly, as does a Gli3, Shh double knockout, but an Shh knockout lacks the digit arch[36]. While our model is unable to show polydactyly, we can perform a double knockout of Shh and Gli3 to see if its downstream effect is more like a Gli3 or Shh single knockout. Our results agree with experimental ones (Fig. 15), although we only have a couple of downstream species to look at. Tissue Mechanics Simulations To test the mechanics of Mech3d, the RD component was turned off, and S (the volume source term) was set equal to a constant. No external constraints were added to the mesh initially. Due to complexities regarding remeshing, this feature was turned off to simplify our first attempt using Mech3D. Time steps between and 0.1 hours were tested. The mechanical parameters used in the model were those that were previously used in tumor simulations. All other parameters are found in the data files in the appendix. The original mesh, before growth, is shown in Figure 16. After several iterations, the limb successfully grew to the shape found in Figure 17. Growth indeed appeared uniform, as the limb scaled up in all directions. Note particularly that the boundary at the flank was enlarged. Von Mises stress was calculated and is shown on the flank surface of the limb at the end of growth (Figure 18). Note that there are only minor stresses present due to the geometry of the mesh. The time step that allowed for the greatest growth was 0.1 hours. At this time step, 8 iterations were completed before the solver was not able to continue. The reason for non-convergence was the presence of elements with negative Jacobians, implying that the elements had collapsed or inverted. Several modifications were made to extend growth, but all were unsuccessful. Several of the future directions in the discussion section address this issue. After some successful uniform growth, the RD equations were coupled with the mechanics via the source term. S was made to be a linear function of FGF4 concentration. In addition, more realistic boundary constraints were created. We fixed the flank surface in all directions (X, Y, and Z). Please see supplemental movie 4. 10

11 Simulations of the mechanics coupled with the RD equations were also successful out to 8 time steps. The growth showed some fascinating features. First, the growth is more pronounced at the AER, starting to form a paddle shape. With the new constraints, the limb did not expand at the flank, but forced most of the growth to occur in the distal direction. Finally, the constraints did cause higher Von Mises stress at the flank surface, localized both dorsally and ventrally. Discussion One of our project aims was to test the accuracy and stability of a reactiondiffusion (R-D) kinetics model by changing the parameters associated with the different morphogens in the system and observing the outcome. An accurate and stable R-D model should give results that mimic those seen in experiments when certain species are perturbed. Our initial tests of our R-D Model through knockout experiments showed that, for the most part, this model functions correctly according to the current morphogen network (Fig. 3). There are a few minor problems that will need to be changed (see results section). However, when compared to knockout and over-expression experiments done in mouse and chick limbs, our model did not always achieve the same results. There could be a number of reasons for the disagreement between the local dynamics R-D model and in vivo experimental results. First, these initial tests of the R-D Model did not account for diffusion of morphogens since it looked at the system occurring in a single cell. When this model gets integrated into the 3D model where diffusion will be active, the results may be in better agreement with the experimental ones. One of the discrepancies was in the Shh, Gre, Fgf4 feedback loop. Shh, Gre, and Fgf4 are all produced in different regions of the limb and diffuse into the other regions to interact in vivo, which is difficult to simulate in a single cell and but could potentially work in a 3D tissue. If the model still doesn t agree once in a 3D system, the equations for each species will have to be reevaluated. Another possible source of disagreement between our model and the in vivo experiments is that the current network for the model is incomplete. The 16 morphogens that are currently in the model are not the only ones known to be involved in limb growth[55]. There are at least a couple of known transcription factors missing from the model. Sp8 and Sp9 are upregulated by Fgf10 and regulate the level of Fgf8 transcription[37]. Knockouts of the transcription factor dhand lead to the absence of the ZPA and Shh production[2]. The model completely ignores Hox genes, although most current research suggests that their role is in digit patterning[17], which is beyond the scope of our current project. The model also ignores FGF receptors which play a role in both A/P and D/V patterning[13]. Only one of the Wnts involved in limb patterning, Wnt7a, is accounted for in our equations. Wnt7a is found at the dorsal ectoderm and regulates Shh, while Wnt3a is needed in AER formation[19]. Wnt3a also 11

12 raises the level of Sp8, which then raises the level of Fgf8 in the limb[37]. The addition of any of these species, or any others known to be active in the limb, into the network may solve some of our discrepancies. We also assumed for this model that no post-translational modifications are taking place, but those modifications may be important for achieving results that agree with experiments. It s also possible that some of the disagreement is due to experiments being performed in mouse limbs rather than chick. While there is a lot of homology between these two systems, there are some known differences. The most obvious difference between the two systems is that mice have five digits while chicks only have three. Thus the digit pre-patterning for the two systems must have some key differences. If these differences reach far enough back from digit specification, then they could be the reason for discrepancies between our model and in vivo mouse experiments. Coupling the RD equations with mechanics showed that the AER and ZPA can cause certain morphological characteristics of the limb. Both the paddle shape of the limb and the dip towards the ZPA reported in the literature were beginning to form in our simulations. There were, unfortunately, problems faced with the mechanics that prevented us from growing a full limb. We have traced the cause (negative Jacobians) to the flank surface of the limb, which is also where the highest stress is seen. It is likely that the stress gradients at the flank surface are causing instabilities in Mech3D. Stresses are localized both on the dorsal and ventral side of the flank surface. The cause of this localization is expected to be from the AER, which wraps around the limb bud from the dorsal to ventral surface. At the boundary where the AER and the flank surface meet, high concentrations of FGF4 are close to the displacement constraints. Rapid growth next to this constrained surface could be the cause of localized stress. In order to rectify this, an additional constraint was added to the source term S. S was constrained such that an element would not grow if its Von Mises stress was above a predetermined threshold, even if FGF4 was present. This not only aids in preventing high stresses, but may be more indicative of the biological system. In tumors, for example, high stresses inhibit cell growth and may even cause apoptosis. This approach is still under testing. Future directions for the mechanics include modifying the mesh and boundary conditions to prevent high stress areas. A new mesh has been developed in which the AER only spans a short arc along the tip of the limb bud, perhaps preventing growth near the flank boundary. In addition, a mesh in which the AER is simply attached on the surface of a very large sphere or cube is being created. This would mimic the growth of the limb from the flank, and would be more similar to the biological process. It would also prevent sharp boundaries and high stress concentrations. More appropriate mechanical parameters specific to embryonic tissue have been found[74], and they will be implemented in future models. Finally, remeshing may solve many problems with the mechanics. Simulations with remeshing have been attempted, but there is still more work to be done in order for this feature to work properly. 12

13 * For supporting online material, please visit: 13

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18 Figure 1: Relevant developmental axes 18

19 Figure 2: Spring and dashpot viscoelastic model 19

20 Figure 3: Network of the 16 genes and gene products accounted for in the R-D Model. F10=Fgf10, F8=Fgf8, F4=Fgf4, Fz=ZPA, S=Shh, P=Ptc, SP=Shh-Ptc complex, SPM=Shh-Ptc-Smo complex, MP=Smo-Ptc complex, Ma=Smo(active), Mi=Smo(inactive), G3a=Gli3a, G3r=Gli3r, Gre=Gremlin, W=Wnt7a. 20

21 Figure 4: Mech3d code schematic 21

22 Figure 5: Relevant tissue mechanics equations Figure 6: Local dynamics with no knockouts, run for 10 hours. 22

23 Figure 7: Local dynamics with no knockouts, run for 1 hour. Figure 8: Fgf10 knockout lowers Fgf8 level. 23

24 Figure 9: Fgf8 knockout (top set of graphs) does not change Shh or the rest of the network. Wnt knockout (bottom set of graphs) changes Shh level, affecting the rest of the network. 24

25 Figure 10: Knocking out Fgf4, Fgf8, Fgf10 and Wnt (top) is the same as turning off the ZPA (bottom). 25

26 Figure 11: Knockouts of Gli3r (top) and Gremlin (bottom). Figure 12: No Shh should be present when the ZPA is turned off, but an initial concentration of Shh-Ptc-Smo causes a small spike in Shh within the first hour. 26

27 Figure 13: Shh knockout lowers Fgf4 and Gre, but not Fgf8. Figure 14: Fgf4 Knockout does not change Fgf10, Fgf8, Shh, or Bmp. 27

28 Figure 15: Shh and Gli3 double knockout (top) gives same downstream results as Gli3 knockout (bottom) not Shh knockout. 28

29 Figure 16: Unconstrained limb bud at time 0. 29

30 Figure 17: Uniform growth caused ballooned limb 30

31 Figure 18: Unconstrained limb bud with Von Mises stress shown. 31