Live Imaging and Computational Modelling of Tissue Growth and Tissue Regeneration in Drosophila
Jamie Rickman
19th January 2015
Abstract Wound healing in the epithelium of the Drosophila wing imaginal disc is a regenerative process which is known to proceed via the assembly of a contractile actin cable that circumscribes the wound drawing the leading edge cells together in a characterstic rosette formation. Here we computation- ally explore various other biological and mechanical processes that are involved in tissue repair in Drosophila; the formation of dynamic actin protrusions that pull each other forward to close the wound; the active extrusion of dead cells from the wound site; and the effect of a global tension force operating on the tissue. In silico simulation results, using the vertex model of epithelial tissue, and in vivo data are compared. It is found that patterning of line tensions in wounded cells and cells neighbouring the wound can drive rosette formation and wound closure. Results from simulating a global tension force indicate this could recapitulate the initial expansion phase of wound healing seen in experiment.
Contents
1 Introduction 2 1.1 Tissue regeneration ...... 2 1.2 Regenerative healing in Drosophila wing imaginal discs ...... 2 1.3 The mechanical drivers of epithelial wound healing in Drosophila ...... 3
2 The Vertex Model 4
3 Current work and results 5 3.1 Wound closure in vivo ...... 5 3.1.1 Wound closure rate ...... 5 3.1.2 Rosette formation ...... 5 3.1.3 Inital expansion of wound following ablation ...... 6 3.2 Wound Closure in silico ...... 7 3.2.1 Implementation of the vertex model ...... 7 3.2.2 Simulating actin protrusions ...... 8 3.2.3 Simulating extrusion of dying cells from the wound site using an equilibrium dy- namics approach ...... 9 3.2.4 Simulating a global tension force ...... 10
4 Discussion 11
5 Acknowlegdgements 12
1 1 Introduction
1.1 Tissue regeneration Tissue regeneration is a remarkable phenomenon and is as yet little understood. Immediately following tissue damage a variety of complex biological systems are activated that can repair a wound with no loss of function. This process requires precise synchronisation of complex intercellular and intracellular pathways that instigate dramatic changes; in cell phenotype, gene expression, differentiation and prolifer- ation. However there are stark differences in the capacity for regeneration across the animal kingdom and this remains a challenging question in developmental biology. Why is that a salamander can regenerate complex body parts and man cannot? Even more puzzling is the question of why a human foetus can perform this kind of regeneration but we lose the ability as adults. Non-regenerative wound repair often results in a mass of fibrotic tissue (a scar) and this poses a huge clinical burden. For example myocardial scar tissue caused by heart attacks is thought to contribute to congestive heart failure and arrhythmia and toxin-induced scarring in the liver is thought to lead to cirrhosis [1].
The ultimate aim of this field of study is to instingate the signalling pathways in adult human tissues that can lead to regeneration. Advances in nanotechnology and bioengineering bring this paradigm closer than ever, in which the biological microenvironment of a wound can be controlled at a patients bedside. Unravelling the coupling between biology and mechanics in wound healing is a necessary first step to understanding the molecular and cellular processes at play. The aim of this report is to investigate a computational model of wound healing based on the vertex model. A biological and physical under- standing will be bought to bear on the model and simulation results will be discussed and compared to experimental data.
1.2 Regenerative healing in Drosophila wing imaginal discs Drosophila is a holometabolous insect, undergoing four life stages from embryo to larva to pupa to adult. The dramatic metamorphoses that take place during these transitions make it a particularly interesting case for studying cell and tissue development. Drosophila can regenerate its tissue in a number of circumstances and is used here as a model organisn. This report looks in particular at regeneration of the wing imaginal disc. Discussion of the complex chemical signalling pathways that orchestrate wound healing in Drosophila lie outside the scope of this report [2],[3]. Here we will focus on the mechanotransduction of these chemical signals, the intrinsic mechanics of the wounded tissue and how the forces that result mediate regeneration.
Following T. H. Morgan’s definitions, a distinction must first be made between the morphallaxis of the Drosophila case and the epimorphosis of the salamander case. The former process is a remodelling of existing tissue to rebuild the tissue architecture while the latter involves the dedifferentiation of a cluster of mature cells in the wound environment that form a blastema which then redifferentiates and proliferates to replace missing tissue. While the epimorphosis of the salamander might seem a more relevant model system since it pertains to vertebrates, it has been found that the mechanisms involved in the morphollaxis of Drosophila are highly conserved across phylogeny. In both Drosophila wound repair and mammalian re-epithelialization the JUN amino terminal kinase signalling pathways are deployed [1]. And in embryonic wound repair, formation of an actomyosin cable and extension of actin structures (filopodia and lamellipodia) are seen in both cases [4]. This has motivated the use of Drosophila as a model system.
The imaginal discs of the Drosophila have received particular attention [5], these are small clusters of cells in the insect larva that will develop into the integument and appendages of the adult body during pupation. They have a number of features that make them amenable to study on a genetic and physical basis;
• Discs are composed of a single layer of columnar epithelium, with a peripodial epithelium contin- uous with the disc at its edges composed of flat squamous cells, see figure 1. Their cells remain undifferentiated until metamorphosis; this makes them easy to image. • Surgery and culture of imaginal discs is relatively simple.
• Because of Drosophila’s well known genetics the system is genetically tractable.
Figure 1: Left: map of the wing disc showing anterior-posterior (AP) and dorsal-ventral (DV) compartment boundaries. Right: The three cell layers in the wing disc: the squamous epithelium or peripodial membrane, the columnar epithelium. Picture taken from [6].
1.3 The mechanical drivers of epithelial wound healing in Drosophila Studies of wound healing in Drosophila have shown that there are two distinct mechanical processes that occur during tissue regeneration. 1. Localization of actomyosin resulting in the formation of an actin cable that circles the leading edge of the wound and acts like a ‘purse string’ to pull the wound closed [7][8][9].
2. Cells at the leading edge of the wound form actin protrusions, namely lamellipodia and filopodia. These protrusions tug on one another to pull cells at the leading edge forward. These two processes work can work in conjunction, with the balance between them determined by factors such as wound topology. For example it has been proposed that larger wounds close primarily by the action of lamellipodia whereas smaller wounds close via the actin cable mechanism. Conversely for small incisions actin cable assembly does not occur and opposing edges simply zip together via lamellipodial protrusions [8].
Following the work of Abreu et al., wound closing will be discussed as a four-stage process of 1) expansion, 2) coalescence, 3) contraction and 4) closure.
The coalescence stage of closure occurs before the actin cable assembles (minutes after wounding), here wound size is relatively stable. Since the coalescence is too short for the up-regulation of actin to have occurred it is thought that re-organization of filamentous actin present in the cell or the polymerization of actin monomer takes place [10]. Intercellular segments of the cable are linked through adherens junc- tions between abutting cells. During the contraction phase the tightening of the actin cable leads to the formation of a characteristic rosette pattern, elongating cells at the wound edge and shortening their linked edges. As the wound tightens some cells at the leading edge are pushed backwards so the wound circumference decreases [8]. The increased cortical tension at these edges has been established through laser ablation experiments. A positive feedback loop, whereby increased tension is sustained and gener- ated by actomyosin localization has been suggested [7]. Previous work from this lab [11] investigating the upregulation of actomyosin found that this process has a characteristic time course, see figure 9b. Lamellipodial protrusions in the apical layer of leading edge cells have been observed during wound heal- ing in Drosophila. Although lamellipodial crawling, whereby leading edge cells pull themselves forward using these protrusions, has been more commonly associated with adult tissue [3], a combination of both this and the actin cable mechanism have been reported in the wing imaginal disc [12]. In the final closure stage both dynamic filopodia and lamellipodia have been observed to protrude from the leading edge cells and perform the final knitting together of the wound [13][8].
Gene knockout experiments have been performed to investigate the action of each mechanism in isola- tion. Mutant Drosophila embryos for Cdc42, a GTPase that regulates formation of actin protrusions, have been shown to be unable to perform the final closing stage, wounds are seen to stabilize. Conversely mutants for Rho1 (necessary for actin cable formation) are able to fully close the wound via actin pro- trusions, with a rate of contraction similar to wild type. The coalescence phase is much longer however, this could be due to the disorganised leading edge not being brought into line by the contractile action of the actin cable. In this scenario it is thought that lamellipodial formation increases threefold, revealing a redundancy in the system [8].
The molecular basis for phase 1 and the expansion of the wound initially after ablation has been inves- tigated using mutant tissues [9]. No mutation was found to significantly affect this phase of wounding suggesting that the expansion is a result of a mechanical property inherent to the tissue. This will be explored further in §3.2.4.
2 The Vertex Model
The vertex model introduced by Farhadifar et al. will be used to for this system; the packing geometry of cells in the wing disc epithelium is well represented by connecting vertices with straight lines to create polygonal cells. Although information is lost by reducing the dimensionality of the system from three to two, it is argued that the surface area of the cells in the vertex model model adequately captures volume information below and since the wing disk epithelium is a monolayer it is a natural simplification.
The vertex model describes a cell junctional network by the energy equation;
X Kα 2 X X Tα E(R ) = α A − A0 + Λ l + L2 (1) i 2 α α ij ij 2 α
The original work extracted normalised parameters for the model from simulations of typical tissue mor- phologies and comparing laser ablation experiments with in vivo data. It was found that the line tension Λij was small and positive meaning that cells boundaries have a tendency to shorten and contractility between cells dominates over adhesion.
The vertex model is plastic in the sense that junctional remodelling can occur, described by T1 and T2 transitions, see figure 2, which are significant processes in wound healing. The rate of T1 and T2 transitions is sensitive to parameter changes and has been used to establish parameter values for the in the original work, this feature of wound closure is not explored here but is an avenue for future work.
Although the ground states of this model are of little biologically relevance, it is informative to note that within the range of parameter values identified by Farhadifar et al. for the Drosophila wing disc epithe- lium; a uniform hexagonal packing geometry is the lowest energy. Introducing cell apoptosis following wounding induces disorder and local perturbations which generate more realistic tissue morphologies. k
Ftension tension F = Λij l i i ∑ α jlk area K F F = α A − A0 l area α 2 ( α α ) jk
j
Figure 2: T1 transitions occurs when a cell is pushed Figure 3: Force diagram for one vertex showing the back from the leading edge; a cell junction shrinks and tension contribution from all connected vertices and then expands in the opposite direction changing the the elastic contribution from one associated cell. Since topology of the network. T2 transitions occurs when the tensional force will have a resultant acting inwards the area of a cell shrinks to zero and is replaced by a on one of the associated cells the elastic force must act vertex, modelling extrusion of dead cells. Figure taken outwards to counter balance this. from [14].
The system then evolves quasi-statically between local energy minima that lie above the true ground state.
3 Current work and results
3.1 Wound closure in vivo In order to establish the key features of wound closure that the simulation had to recapitulate, analysis of laser ablation experimental data obtained by this lab was undertaken in conjunction with a review of literature. The key features of wound closure were identified as;
1. Characteristic phases of wound healing with associated rates of closure. Following the work of Abreu et al., wound closing will be discussed as a four-stage process of 1) expansion, 2) coalescence, 3) contraction and 4) closure.
2. The formation of a characteristic rosette pattern around the wound.
Transgenic fly stocks expressing DE-cadherin GFP and Sqh-mCherry were used for the laser ablation experiments. This allowed imaging of E-cadherin (a cell-cell adhesion molecule) to image cell borders and regulatory myosin light chain to image actin cable assembly. Analysis was performed using ImageJ and Matlab (data courtesy of Rob Tetley, Ross Harper and Yanlan Mao).
3.1.1 Wound closure rate Two wounding experiments were analysed with wounds of different sizes to explore the effect of wound topology on healing. With an inital wounding of approximately 30 µm (wound A) expansion of ∼ %15 is seen. There follows a long phase of coalescence and then a rapid contraction phase, see figure 4. In marked contrast, with a wound of approximately 80 µm (wound B) there is an initial expansion of ∼ %20 but no coalescence phase, see figure 5. The differences in phasic behaviour suggests that wound topology plays a role in the determining the mechanisms by which healing proceeds, however the size of the wound is probably one of many factors at work, and some report that wound size plays no role at all, citing experimental procedures as the most likely determinant [9]. However in both cases the contraction phase is well fit with an exponential, see figure 7, in the case of the smaller wound a two term exponential significantly improves the fit. This could imply two rate constants are controlling wound closure [15], however more accurate analysis is required before conclusions can be drawn.
3.1.2 Rosette formation In order to characterise the formation of the rosette of leading edge cells two features were explored to use as morphological benchmarks. It can be seen that the cells forming the rosette both elongate and Figure 4: Wound A with an initial area of 30 µm. Figure 5: Wound B with an initial area of 80 µm. The wound increases by 15% during expansion. No The wound increases by 20% during expansion, this is coalescence is seen and the rapid contraction phase is followed by a long coalescence and then a rapid con- well fit with a double exponential. traction well fit by a singly exponential.
Figure 6: Elongation ration during closure of wound A. Figure 7: The contraction phases of wound closure for An initial steady period mirrors the coalesence phase, A (blue line) and B (orange line) are best fit by a followed by an increase during contraction and a final double and single exponential respectively. drop as the cells rearrange. reorient as the wound closes, aligning their major axis with the centre of the wound. The elongation ratio (ER) of the best fitting ellipse (calculated with the least squares method in Matlab) was therefore majoraxis obtained to quantify the amount of elongation given by minoraxis . Reorientation of the cells was mea- sured by calculating the angular displacement between the major axis of the best fitting ellipse and the radial axis joining the centre of mass area of the wound and the cell. Figure 6, pertaining to wound A shows the coupling of wound size and ER. As the cells elongate they fill the wound and thus the same period of coalescence is seen, followed by a rapid increase. In the final closing stage the ER drops as rosette formation disassembles and junctional remodelling takes place to facilitate closure. Analysis of orientation did not show robust trends, there are several possible reasons for this. Orientation is not a characteristic property of a cell unless it has a relatively high ER, making orientation redundant in the initial and final stages of closure. In addition imperfect rosette formation and elliptically shaped wounds also diminish the value of this measure. We will proceed to use ER only as a benchmark and defer discussion to §4.
3.1.3 Inital expansion of wound following ablation There are two possible explanations for the expansion phase seen in the experimental data. Either the tissue holds individual cells above their preferred area, such that immediately after ablation cells neighbouring the wound cells shrink back from the wound edge as the line tensions at the border are broken. Or a global tension is operating on the tissue and is stretching the cells, (which are being held below their preferred area) such that a rupture in the tissue causes an initial recoil. As a preliminary investigation into which of these two scenarios pertains, the area of cells around a wound were measured before and after ablation. It was found that there was on average an approximate 4% increase in cell area, implying that the cells are in a state of compression and that the latter case pertains. Furthermore investigation into a molecular basis for expansion using mutant tissues yielded no results [9], again suggesting that it is a result of a mechanical property inherent to the tissue. Image analysis of cells in the second and third row from the wound should corroborate this theory and greater accuracy in measurements here should be a goal for future work.
3.2 Wound Closure in silico 3.2.1 Implementation of the vertex model A simulation implementing the vertex model developed in C++ by this lab was used for the following work. Parameter values used also follow previous work from this lab [16]. Two modifications to the basic vertex model are implemented; 1. The third perimeter contractility term is not used as it is found to be redundant in certain parameter ranges where the tension term can fully account for the behaviour of the system.
2. To simulate the action of actomyosin cable formation a time-varying line tension, Tborder has been included at the border between healthy and ablated cells, see figure 11. Results for the time course and parameter space for this feature will be taken from previous work by Ross Harper. 1.
The system described by equation (1) is evolved by modelling the tissue as an overdamped frictional system and solving equation (2) at each time step to calculate the velocities of vertices using (2). dE(R) dR = −γ (2) dR dt In order to simulate the wounding it was necessary to decide upon biologically relevant parameter values for the ablated cells. After wounding, the ablated cells lose their structural integrity and no longer form a cohesive tissue which motivates setting the line tension of ablated cells to be zero and follows Farhadifar et al. However they remain as debris at the wound site and resist compression, which motivates keeping AB the elasticity coefficient finite. It was chosen conservatively that Kα = Kα since some studies also report an increase in cell elasticity during apoptosis [17]. Determining the preferred area of ablated cells is discussed in section 3.2.3.
The following results describe three modifications made to the simulation:
1. Simulating actin cable formation without the effect of actin protrusions means that full wound closure is not achievable in a large region of parameter space, wounds instead have a tendency to stabilize at a finite area [11]. Section 3.2.2 seeks to address this issue.
2. Determining the preferred area of ablated cells poses subtle problems. The cells must resist com- pression somewhat but also allow themselves to be extruded from the wound site, which is itself an active process integral to wound repair [18]. Section 3.2.3 offers one solution to this problem.
3. The initial expansion phase that is seen in vivo cannot be captured with the forces modelled by equation (1) and the parameters values as above. We note that at steady state the simulated cells are held below their preferred area and therefore the elastic energy term produces a force acting radially outwards from each cell, see figure 3, causing wound ingression not expansion. Section 4 explores adding a global tension term to the system to recapitulate the expansion phase.
1Parameters for the time course of this process were obtained through analysis of fluorescence microscopy images using pixel intensity as a measure for relative myosin concentration and assuming a linear relationship between line tension and myosin concentration. 3.2.2 Simulating actin protrusions In simulation results from previous work the wound area is seen to become stable for large regions of parameter space, rather than close completely [11]. Figure 9b shows that the myosin concentration drops to baseline levels before the complete closure of the wound, indicating that another mechanism may be responsible for the final closure stage. Qualitatively we can understand that as the rosette of cells around the wound tightens and the cells become more elongated they will begin to resist further deformation through contraction of their leading edge. Another mechanism is needed to instigate the final remodelling of junctional contacts into a more relaxed configuration. It is here proposed that actin protrusions play a significant role.
This mechanism was simulated in the model by adding time dependent line tensions on the in- Figure 8: Parameter space of Tinner and Tborder showing time for side edges of leading edge cells, see figure 11. wound closure. Red indicates wound stabilizes without closure. The time course was chosen to have the same delayed onset as the actomyosin upregulation to allow for the formation of the actin struc- tures on a biologically realistic time scale. The lowered line tensions were then held constant since studies report this mechanism to be ac- tive in all stages of closure [8]. The cell junc- tions held under Tinner would therefore have a tendency to elongate relative to their neigh- bours. Simulations were run varying this min- imum tension in the biologically realistic range 0 < Tinner < Te and Te < Tborder < 2Te where 2 Te is the tension of healthy cells , the param- eter space of wound closure is shown in figure 8.
(a) (b)
Figure 9: (a) Simulation results for wound closure showing the effects of varying Tinner and Tborder in isolation. Lowering only Tinner can close the wound, whereas increasing only Tborder cannot. (b) Myosin upregulation begins several minutes after wound closure and quickly reaches a maximum value. The response begins to drop at 1200s and returns to baseline at which point the contraction of the wound ends. Parameter values for blue line in (a).
Figures 9a shows the simulated action of the actin protrusions and the actin cable in isolation. It can be seen that both lowering Tinner (simulated actin protrusions) and raising Tborder (simulated actin cable) increases the rate of contraction. However increasing Tborder is not sufficient on its own to fully close the wound, even when increased to double its normal value the wound stabilizes at ∼ 20% of its original size. Figure 9b shows that once Tborder falls back to baseline the contraction phase comes to an end and final closure of the wound cannot take place. However wound closure can be achieved by just lowering Tinner. The results shown in figure 9a are coherent with the gene knockout experiments of section 1.3; 2Simulations were run with the equilibrium area condition given in section 3.2.3. the actin cable cannot close the wound alone but filopodial and lamellipodial action can. Not too much credence should be given to this result until biological evidence for the simulated time course of the actin protrusions is found, however it does support the in silico implementation of these two wound-healing mechanisms as temporal patterning of junctional tensions around the wound edge.
Comparison of 9a with in vivo results shows more similarity with the closure of the larger wound. Expansion is not seen, this is addressed in section 3.2.4 and there is no significant coalescence phase. It has been suggested that coalescence, rather than being a passive process of contractile forces overcoming wound expansion, it itself an active process with its own molecular mechanism and is therefore context dependent [9], explaining why it is not seen in silico. Choosing median parameters with the feasible T e ranges given above to proceed with (Tinner = 2 ,Tborder = 0.75T e), the contraction phase is well fitted with a double exponential, compatible with experimental data and other studies [9].
Figure 10: ER as a function of time. Lowering Tinner results in elongation of cells and the formation of the characterstic rosette (blue line). The signature dip at Figure 11: Schematic of patterning of tensions at cell final closure is also recapitulated here. ER remains junctions around the wound edge. constant as wound size plateaus if Tinner is not re- duced (orange line).
Rosette formation was assessed by analysis of ER. Figure 10 shows that lowering Tinner results in a steeper increase in ER and also captures the sharp drop at final closure as the cells rapidly rearrange, as seen in vivo. Without lowering Tinner the ER plateaus with wound size as expected.
3.2.3 Simulating extrusion of dying cells from the wound site using an equilibrium dy- namics approach In the previous work [11] the preferred area of the ablated cells was held equal to that of healthy cells but their elastic coefficient Kα was reduced, allowing them to be extruded from the epithelial layer by decreasing their resistance to compression. However Kα had to be dropped to 30% of the standard value before full wound closure could be established. Since the dead cells remain in the wound matrix the resistance presented by them would not be insignificant and only a small reduction in Kα seems justified. An alternative approach is an equilibrium dynamics method, used by Nagai et al. In this scenario we AB keep Kα = Kα but the preferred area of the ablated cells is coupled to the size of the wound through 0 the equilibrium condition given by (3). AAB is the preferred area of an ablated cell, n is the number of ablated cells and Sn is the wound area. As the wound shrinks the preferred area of the ablated cells gets smaller mimicking their extrusion from the epithelial layer. Although there are conceptual pitfalls 0 to this method (some cells will necessarily be below AAB and will experience an expansion force which might not be biologically feasible), it allows the extrusion of dead cells to be a dynamic process internally maintained by the whole tissue system. S P A A0 = = n n (3) AB n n
Choosing median parameter values for Tborder and Tinner within the feasible range with which to proceed, figure 12 shows the different phasic behaviour of wound closure with the two methods. If the preferred area of ablated cells is held constant the onset of the contraction phase is rapid and proceeds at constant rate best fitted with a single exponential. However using an equilibrium dynamics approach two stages of closure can be seen. Initially the wounded cells present high resistance to compression but as the wound tightens their preferred area decreases relative to the surrounding cells. This results in a slow contraction followed by a more rapid contraction which is best fit with a double exponential. Further statistical analysis of both simulated and experimental data is needed here to draw conclusions as to which method improves the model.
Figure 12: Keeping A0 constant results in a contrac- Figure 13: With a stretching force applied radially tion phase fitted by a single exponential. Using the outwards from the centre of the tissue an expansion equilibrium area condition for A0 results in two stages phase is seen giving a 2% increase in wound area. 5% of contraction fitted by a double exponential. Note the stretching refers to the increase in area of cells in the initial expansion seen in the former case is an artefact central perimeter defined by radius a (see text). of reducing Kα.
3.2.4 Simulating a global tension force In order to recapitulate in silico the initial expansion phase seen in vivo it is proposed here that a global tension force operates on the tissue perhaps due to friction from the substrate beneath or being stretched by the peripodial membrane. In order to simulate this force another energy term, UG was added to equa- tion (1) given by (4). The form of this potential is such that no force acts on cells within radius a of the centre of the tissue and cells at the edges experience the maximum force which propagates through the tissue to stretch the central cells.
X 2 UG = β (r0) r0 (4) i
0 r ≤ a β(roi) = σ/2 a < r ≤ b σmax/r0 r > b
Index i sums over all vertices and r0 is the distance between the centre of the tissue and vertex i. Clearly there are limitations to modelling the potential in this way. The wing disc is not symmetric about the z axis as this potential would imply. In implementing this model a square tissue was used with a modified square potential. In figure 14b it can be seen that this square potential stretched the tissue preferentially in some directions making it unstable to large tensional forces. It was sufficient however to stably stretch the simulated tissue, within the perimeter defined defined by a, to +5 % of its normal value. Figure 13 shows that with this additional force an initial expansion of the wound is seen, followed by a normal contraction phase. The expansion of the wound size is only +2 %, far smaller than experimental data. A higher stretching force applied with an improved potential could provide better results and this is an avenue for further work.
(a) (b) Figure 14: (a) The maximum recoil of vertices of cells around the wound edge is strongly dependent on position with a sharp peak seen in the direction of stretching. A peak at 30◦is seen here. (b) Diagram showing wounded cells (green) and the stretching force acting radially from the centre of the tissue.
Further analysis of this expansion phase shows that the recoil of vertices around the wound edge is dependent on the positioning of the wound in the tissue. Figure 14b shows a wound in the top right corner of a square tissue and 14a shows the maximum displacement of vertices of cells surrounding the wound during the initial expansion phase. A sharp peak is seen at around 30◦ showing that cells lying along a radial axis, i.e. the direction of stretching, experience a greater recoil. This result demonstrates a feature of the simulated force that could be compared to experimental data both to refine the model and probe the real form of the potential in the tissue. Further information could be gathered here by looking at the direction as well as magnitude of recoil of the vertices.
4 Discussion
This report has looked at a simulation of the wing disc epithelium of Drosophila using the Vertex model described by Farhadifar et al. and compared results to experimental data of laser ablations and a review of literature. Analysis of in vivo data revealed phasic behaviour of wound closure that was markedly different between experiments. It was proposed that wound topology and context could determine which closure mechanisms predominate and the relative time periods of different phases of healing. This is an avenue for future research requiring more experimental data and better analysis tools. In addition exploration of a molecular basis for coalescence should be undertaken in order to ascertain whether it needs explicit implementation in a computational model of healing. In both experimental cases analysed the contraction phase is an exponential decay in wound size. Rosette formation can be characterised by an ER which inversely correlates with wound size, and a dip in the final stage of closure. A better char- acterisation of rosette formation to include orientation of leading edge cells could improve comparisons, for example orientation could be used as a multiplicative factor (90◦→ 0 and 0◦→ 1) to scale ER and penalise cells that are out of alignment.
Simulations involving only actin cable formation cannot robustly recapitulate wound healing. In partic- ular the elasticity of ablated cells, given by Kα has to be dropped to ∼30% of its normal value to achieve full closure. In light of this a second mechanism for wound closure was introduced; actin protrusions that tug at each other on the leading edge and pull each other forward. Wound closure was then achievable AB in silico across a much larger region of parameter space with Kα = Kα, which is here proposed to be a more biologically feasible value and is in keeping with recent work [18]. Furthermore this introduced a redundancy in the system; wound closure could be achieved with no simulated effects of an actomyosin cable, also in correlation with literature [8],[9]. Experimental data to quantify the role of actin protrusions would improve this work, live actin dynamics can be visualised in epithelial cells expressing a GFP actin fusion protein. This would help establish a time course for actin protrusions and any feedback systems present, such as the upregulation of actin protrusion formations when actin cable assembly is inhibited [8].
Following Nagai et al. a new approach was taken to model extrusion of cells from the wound site. Rather than keep the preferred area of cell constant it was taken to be the average size of cells in the wound. The decrease is wound size in the former case is best fit with a single exponential decay and in the latter, a double exponential decay. Further analysis of experimental data will have to be undertaken to draw a firm conclusion on the merits of this new method.
Finally a global tension force was added to the simulation. It is here proposed that the expansion phase seen immediately after wounding is a result of inherent mechanical tension in the tissue. A better force potential to model this is required to simulate higher tensional forces, simulation results show expansions far smaller than experiment. The physical consequences of this global tension were explored in silico and it is found that angular displacement of vertices around the wound edge reveals information on the form of potential via positioning of the wound.
The biological mechanisms behind wound healing are complex and remarkable. By investigating various implementations of these mechanisms in silico, with comparison to experimental data, we can hope to gain a better understanding of the interplay between the different drivers at work. Ultimately this will push forward the field of regenerative medicine and get us closer to developing bio-engineering techniques for clinical use.
5 Acknowlegdgements
I would like to thank Yanlan Mao for her supervision with this work and Rob Tetley and Melda Tozluogo for their help and advice.
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