Deterministic outcomes from inherently stochastic components

Biological systems are intrinsically noisy but nonetheless produce deterministic outcomes. During development, organisms utilize signaling molecules, i.e. morphogens, to generate a body plan and differentiate cells. With modern experimental techniques, it is possible to measure temporal concentrations of selected morphogens in each cell. The expression of a gene depends on the probabilistic outcomes of several factors, such as molecular binding affinities, processivity, and regulatory sequence interactions. Specifically, genetically identical cells produce morphogen transcripts in asynchronous bursts and in varying quantities [1]. It is therefore essential to investigate how ordered deterministic structures are formed from underlying stochastic components, such as gene expression. In this study, as an approximation to the stochastic nature of morphogenesis, parametric variation is introduced in the partial differential equations of the mathematical model. Earlier studies have addressed robustness of developmental processes, also termed as canalization [2], which remain a subject of great interest today [3–6]. Developmental robustness has been highlighted as a necessary condition that narrows down dramatically the search for plausible models and one that can even “predict key mechanistic and molecular properties of the associated biochemical circuits” [3]. For example, the Turing mechanism has been deemed inapplicable in many instances due to its sensitivity to noise common in developmental processes [6]. In this paper, we analyze the response of a developmental system to noise quantitatively from a statistical physics perspective.

Recent studies in developmental biology have addressed the issue of robustness [7, 8]. For a model to be plausible, there are requirements that the suggested mechanism is robust when parametric noise is introduced. However, in previous works the definition of robustness has mostly been used in a qualitative manner, and the detailed properties of the disordering mechanisms have been left unexplored. In this contribution, novel quantitative measures of robustness are developed and, the value of their application is demonstrated in the analysis of pattern formation in finite-system simulations. These new tools enable the determination of the functional form of the model’s response to stochasticity, leading to a rigorous description of a noise threshold. Specifically, a low-noise regime of robustness is identified in which, even in the presence of stochastic input, there is a deterministic outcome, i.e. structural order. In addition, these new quantitative measures of structural order allow for a precise analysis of the behavior of the system after this threshold is crossed. This disordered regime is relevant to genetic mutations, environmental perturbations and cryptic genetic variation.

Pattern formation in the Drosophila eye disc is an ideal model system to study the effects of stochasticity on deterministic development

The compound eye, found primarily in insects, consists of an array of repeating visual units. The Drosophila compound eye is made up of approximately 800 unit eyes, known as ommatidia. In wild-type Drosophila, the structure of the ommatidia resembles a near-perfect hexagonal lattice. This highly structured pattern is developed via the delicate coordination of cell signaling, proliferation, movement and apoptosis [9]. Some of these cellular processes are guided through the communication of a few conserved molecules, known as morphogens, resulting in tissue morphogenesis.

While the resulting functional eye emerges in the adult, the role of each cell within repeating ommatidial arrays, and other head structures, are specified during larval development [9]. The larval eye-antenna imaginal disc, hereafter referred to as the eye disc, contains numerous cells that produce various morphogens. The Drosophila eye disc is one of the simplest systems for studying morphogenesis since this tissue develops in an approximately two-dimensional fashion. This is because the eye-antenna disc is derived from an epithelial monolayer [10]. Furthermore, this is an ideal system to model morphogenesis, since cell lineage has a minimal effect on pattern formation, and differentiation relies primarily on cell-to-cell signaling [11]. The underlying developmental mechanisms that guide the formation of the eye disc from larva to adult have been studied extensively, allowing for the generalization and analysis of biologically realistic models.

To investigate the stability of the developmental pathways in the Drosophila eye disc, numerical simulations are used, and the results are analyzed from a physics perspective. The spatial organization of the ommatidia begins with the specification of the R8 cells, the future centers of the ommatidia, on the undifferentiated cells of the eye disc, as shown in Fig 1. Mathematical modeling of the Drosophila eye disc pattern-formation mechanism has been the focus of previous investigations, [7, 8] which proposed a model that reproduces the hexagonal lattice pattern of differentiated R8 cells, which is robust enough to be biologically plausible. Using this latest model as a basis, we examine and quantitatively test the robustness of the emerging spatially ordered patterns of differentiated R8 cells when transcriptional noise is introduced. Until today, only qualitative measures of spatial order of the pattern have been applied in this mathematical framework [7]. To this end, we implement several appropriate measures of spatial order, testing the functional relationship of R8 cell pattern order with increasing stochasticity.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 1. Experimental image of the positions of the R8 cells during morphogenesis.

In this fluorescent microscopy image, one can see DE-Cadherin linked with GFP. This labels the top part of each cell shown as a ring. When multiple cells are in close proximity, they appear as one bright point, which in this case point the positions of the R8 cells.

https://doi.org/10.1371/journal.pone.0210088.g001

During the development of the eye disc, morphogens are produced and emitted by individual cells in quantities that are determined by morphogen inputs from other cells in their vicinity [9]. In addition, the production and diffusion of morphogens is non-uniform, i.e. a given morphogen concentration in a cell does not result in a single production or diffusion rate. Therefore, the highly ordered structure of the ommatidia in the Drosophila eye disc, emerging from such stochastic constituents, requires robustness in the underlying gene regulatory networks. Our study aims to quantify eye disc developmental robustness. We focus on one essential step out of many involved in eye disc development, in which additional robustness is introduced [12]. However, the regulation of R8 cell distances is a crucial developmental step in precisely placing each repeating eye unit, thereby strongly influencing the organism’s overall visual acuity.

Biological framework of R8 cell specification

Each ommatidium is made of 20 cells, 8 of which are photoreceptors. The specification of the first photoreceptor, R8, guides the specification and orientation of the remainder of the cells within the ommatidium [9]. Thus, the spacing between neighboring R8 cells is pivotal for positioning and refining geometric relationships in resulting ommatidia. Cellular defects that misplace a single ommatidium will influence the position of neighboring ommatidia, thus propagating further flaws in the lattice. If even a single gene involved in ommatidia formation is perturbed, all unit eyes can be affected, since the regulatory logic that generates each ommatidium is repeated using the same set of morphogens and gene networks [9].

Prior to R8 cell specification, the eye disc is composed of tightly-packed undifferentiated cells dividing asynchronously. Differentiation starts with the initiation of the morphogenetic furrow (MF), a physical indentation in the eye disc that sweeps through the tissue to dictate the pace at which ommatidia are specified and positioned. The MF advances anteriorly via the communication of several morphogens, primarily Hedgehog (Hh) and Decapentapegic (Dpp). The former morphogens have been identified as the long range diffusible activators for the production of atonal (ato), the transcription factor that initiates R8 cell differentiation [13, 14]. The MF is initiated in the posterior end of the eye disc and sweeps through the tissue towards the anterior end. As the MF sweeps through the eye disc, the cells posterior to the MF differentiate and commit to their respective role within the ommatidium. Anterior to the MF, the cells are asynchronously proliferating but at the anterior interface of the MF, cells are arrested in G1 to allow synchronous divisions [15].

In the anterior part of the MF, the transcription factor atonal is expressed in intermediate levels in a stripe by all cells. As the MF progresses, the stripe of ato-expressing cells converges to clusters of 12 cells and finally, to a single ato-expressing cell, the R8 cell [9, 16]. The inhibition signals originating from the refining ato-expressing clusters define the template for the refinement of ato-expresing stripes to the subsequent column [14, 16]. Based on the results of [7], which deem the role of Scabrous as a short range diffusible activator, the diffusible inhibitor of stripe refinement has not yet been identified but is related to the epidermal growth factor [14]. Finally, the Notch-Delta pathway is known to be associated with both stripe and cluster refinement [14, 17, 18].

Process leading to pattern formation in the Drosophila eye disc

In this section, we provide a simplified description of the rational behind the current mathematical model, describing the basic mechanism of pattern formation in the eye disc. The goal is to provide the reader with an intuitive understanding of the foundations of the mathematical model by introducing the minimum amount of elements and interactions. The details of the full model are elaborated in the methods section below.

The cells of the eye disc are assumed to lie on a hexagonal grid, as appropriate for a hard disk tight packing. After the MF passes, a portion of these cells differentiate to become activated R8 cells, and the remainder will be left undifferentiated. At the end of the process, ato-expressing cells are positioned on a hexagonal lattice. This process is illustrated in Fig 2.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 2. Visualization of the foundations of the R8 cell specification mechanism in the Drosophila eye disc.

Simplified illustration of pattern formation mechanism in the Drosophila eye disc as a result of the competition between short-range inhibitor and long-range activator morphogens. Posterior region of the eye disc with an initial row of differentiated precursor cells, denoted by red dots. The morphogenetic furrow (MF), modeled by a plane wave front, moves to the right towards the anterior region. The gray circles represent the boundaries of regions affected by the short-range inhibitor, where the morphogenetic furrow (MF) will not initiate production of atonal. Therefore differentiation can only occur in those locations which are not affected by the short-range inhibitor, leading to the hexagonal supper-lattice of differentiated R8 cells, which is experimentally observed. This diagram does not incorporate cluster formation.

https://doi.org/10.1371/journal.pone.0210088.g002

Some morphogens are diffusible, while others are cell-specific and stay within the boundaries of the cell. In this study, ato is the only non-diffusing, cell-specific morphogen, while all others can diffuse between cells. The various morphogens either have inhibitory or activating properties. The inhibitors are morphogens that, when present in a cell, decrease the rate of production of another morphogen, and vice-versa for the activators.

The following paragraph defines elements and their interactions in an approximate fashion in order to provide an understanding of the patterning process with minimum complexity. Initially, we will deal with the production of single R8 cells rather than ato expressing clusters, which are defined as cells that express ato after the MF has passed. There are two main ingredients that make the pattern of R8 cells form. First, there is an inductor (or activator) signal that represents the MF, which causes undifferentiated cells to produce atonal. It can be approximated as a rectangular wave form, with one edge expanding with a constant velocity that eventually moves through the entire hexagonal lattice. Second, there is an inhibition signal that blocks atonal when present above a threshold. When a cell expresses ato above a threshold it rapidly, compared to the speed of the MF, produces an inhibitory signal that represses the production rate of ato of all cells within its vicinity, depicted by the circular regions in Fig 2. Ultimately, the first cells that do not lie within the inhibitory regions, extending from the most anterior column of ato-expressing cells, will become R8 cells.

As an initial condition, a periodic array of differentiated cells are placed as in the posterior region. The spacing of these cell clusters is chosen such that it will allow for the pattern to propagate. In this simplified setting, the initial configuration is defined by differentiated cells separated evenly in a single row in the posterior-most region. The MF then starts to propagate from posterior to anterior and causes the pattern to propagate as shown in Fig 2.

The actual model that leads to pattern formation is more elaborate. Now that there is a basic understanding of how the pattern propagates, some more intricate aspects can be introduced. In order for the model to be biologically plausible, cell clusters are formed instead of single cells. Even in this illustrative context, as it can be inferred schematically from Fig 2, the propagation of the pattern is very sensitive to the shape of the inhibitory regions. A recent refinement of the model eliminates this flaw by including a short ranged diffusible activator that is emitted rapidly, compared to the speed of the MF, by ato expressing cells. In simple terms, this forces the first uninhibited cell to receive the linearly propagating activator signal to create a circular activator region. This activator region defines the cluster size and shape and makes the model robust, preventing the creation of greatly elongated cell clusters, as explained in Ref. [7].

Mathematical formulation

We start with a model of coupled differential equations that describe morphogenetic pattern formation for R8 cell specification on the Drosophila eye disc obtained from [7], an article expanding on the work of [8]. The model relies on the simplifying assumption that the number of cells in the eye disc is fixed [7, 8]. The cells in the eye disc are arranged in a hexagonal grid, treating it as an underlying two-dimensional structure. The lattice constants for the hexagonal lattice are set to 1, and the points of the lattice are defined by the relation , where (i, j) are integers and are the perpendicular unit vectors in the x and y directions respectively. Every cell, due to the hexagonal geometry of the lattice, has six adjacent neighbors. The resulting coupled partial differential equations are of the form: (1) (2) (3) (4)

These represent the spatio-temporal evolution of the morphogen concentrations a, u, s, and h, responsible for pattern formation in the drosophila eye disc. The morphogen interactions are summarized in Fig 3(a). Here, the upper case indexes label each cell. The pro-neural transcription factor marking the center of the future ommatidia is a (for atonal), u represents all diffusible inhibitors, s denotes short-range diffusible activator (scabrous), and h (for hedgehog) describes the morphogenetic furrow (MF) which activates production of atonal along a propagating wave front. MF propagation is mathematically derived to have this functional form from the underlying morphogen differential equations [7]. The constant v is the velocity of the MF, D_h is the diffusivity, P_h the production rate, and τ_h is the reaction time scale of hedgehog.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 3. Interactions between morphogens and the simulation of the R8 cell ordered pattern.

a) In this diagram we show the interactions defining the regulatory network between the four morphogens in the mathematical model. h acts solely as an activator to a, a increases production of itself and leads to production of u and s, s activates a and lastly, u inhibits the effect of h and s. b) Ordered R8 cell pattern obtained by the mathematical model. The pattern was obtained numerically using the parameters used in [7] (shown in S1 Table).

https://doi.org/10.1371/journal.pone.0210088.g003

The mathematical model contains multiple parameters, operators and functions that play important roles in the dynamics of the system. First, the characteristic reaction time scale of each morphogen is given by corresponding τ’s on the left hand side of the equations. In addition, each morphogen’s differential equation contains a term with a −λ coefficient that incorporates the spontaneous decay of the morphogens. This term, combined with τ, determines the mean lifetime of the morphogen via the ratio . In addition, for the morphogens that diffuse, there are Laplacian operators associated with diffusion, Δ which are discretized at the cell level (see S1 Appendix for exact expression). For cells on the boundary, reflective boundary conditions are used. Note that although only the equations for u and s have this diffusion term, the differential equations that lead to the functional form of h are also based on diffusion. Each Laplacian operator is multiplied by a term D, the value of which (when divided by the respective τ) sets the scale for how fast the morphogen diffuses.

The Heaviside functions are essential, as they set the morphogen concentration thresholds that start, accelerate and stop production. They are a simplified version of the more biologically plausible sigmoid functions, and are defined by (5)

The Heaviside function θ(hⁱ − h₁) in Eq 1 is responsible for initiating production of atonal, a, in the target cell with a rate , once the concentration of h crosses the threshold h₁. This term is multiplied by (1 − θ(uⁱ − u₁)), a function that goes to zero when uⁱ > u¹. This term introduces the functionality of u as a direct inhibitor of atonal in the model. The last term in Eq 1 introduces s as an activator of atonal. Furthermore, the first term in Eq 1, P_aθ(aⁱ − a_a), sets a threshold for atonal at which its production becomes refractive to inhibitory signals. Finally, the two remaining differential equations for the morphogens u and s in Eqs 2 and 3 have similar functional forms. Here, the Heaviside functions specify the threshold value for atonal in the target cell that is needed for initialization of production.

In this model parameter sets have to be tuned for the experimentally observed ordered pattern to emerge. Furthermore, as the placement of the ato expressing cells can vary in cluster size and separation, we use a biologically plausible and stable parameter set, determined by comparison of numeric simulations to experimental imaging data of developing drosophila ommatidia previous stability analyses adopted from [7, 8], shown in S1 Table. More specifically, the decay length scales and time scales for morphogen concentration levels used in the numerics agree with the known experimental data [8]. In Fig 3(b), we show simulation results for this ideal parameter set after the MF has passed the entire cell grid.

The mathematical model is simulated numerically on a hexagonal lattice of 44 vs 120 cells. The size and shape of the simulation grid does not reflect the actual size of the eye disc. The mathematical model evaluated on the 44 vs 120 simulation grid, while using the parameter set shown in S1 Table, results in the differentiation of 85 ommatidia in contrast with the ∼800 ommatidia present in the wild-type eye disc. Moreover, the simulation output contains 20 columns of cells with alternating four to five cells on each column. In contrast, the real eye disc contains 32-34 columns and the number of ommatidia per column varies along the eye disc to provide its elliptical shape [9]. Therefore, the custom size used in the simulations is acceptable to probe how noise propagates between columns while keeping the system size small enough for computational efficiency.

Introducing noise.

Starting from validated simulations of the noiseless case [7, 8], we are now well positioned to study the effects of parametric disorder on pattern formation. The motivation for this is that even in quasi-identical cells, it is experimentally observed that genes are expressed at appreciably different levels due to stochasticity. Here we investigate how such variations affect the outcome of pattern formation in the Drosophila eye disc. To introduce effects of disorder to the system, one or two parameters of interest are chosen at a time. The reason for this choice is to probe the system response when one of the many components start to fail as often encountered in genetic mutations and other stresses. The analysis for adding equal amount of noise to many parameters and adding variable amounts of noise for a pair of parameters was included for completeness in S2 Fig.

For the chosen model parameters, their value is picked from a normal distribution centered at the mean value that produces the ordered pattern of differentiated R8 clusters in Fig 4(a). For each cell of the system a unique value for the parameter is drawn. They are then kept constant during the time evolution. The width and center of each Gaussian distribution is chosen to be the same for all cells. When more noise is introduced, the width of the distribution is increased accordingly. More specifically, in the numerical data discussed below the widths are tuned from 0% to 60% of the mean. As an example, introducing noise in the production rate of u corresponds to picking values from the probability distribution: (6) where μ = 〈P_u〉. Introducing noise in the diffusivity of the morphogens is a bit more intricate and is explained in S1 Appendix. The effect of introducing noise in the diffusivity of u and s is presented in Fig 4. The interpretation of adding noise to the diffusivities corresponds to cell size variation. In this paper, refers to the standard deviation of the normal distribution, as shown for P_u in Eq 6, and therefore defines the degree of noise. In the unlikely instance that a negative value is drawn from the distribution, the absolute value is considered.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 4. Pattern formation simulation results in the Drosophila eye disc, the morphogenetic furrow moves from left to right.

Cluster positions and sizes of ato expressing are shown for increasing noise σ in the diffusivity of the morphogens D_u and D_s. (a)-(c) are the final patterns for σ/μ = 0, σ/μ = 30%, and σ/μ = 40% respectively. Here, μ is the mean of the corresponding normal distribution and the value that is used to generate the perfect pattern.

https://doi.org/10.1371/journal.pone.0210088.g004

Cluster refinement.

Until now, in the parameter regime deemed biologically plausible, the mathematical model in Eqs 1–4 produces patterns containing clusters. However, the actual developmental process includes an extra step that reduces the clusters to single activated R8 cells, which then become the center of the future ommatidia. This process is associated with the Notch-Delta pathway, whereby the cell that produces the most Delta inhibits its nearest neighbors [18]. In order to reproduce this refinement process, a single cell for each cluster is kept as the R8 cell. The cell is chosen to be the most central cell in the most populated row in each cluster, an approximation to the non-trivial pathways and models identified in the literature [7]. This cluster refinement process is illustrated graphically in Fig 5. This extra step, of cluster refinement, has been analytically shown to increase robustness of the pattern [19].

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 5. Visualization of the cluster refinement step.

The code takes the ato expressing cell clusters, as shown in (a), as an input and determines the single R8 cell that will become the center of the future ommatidium, as shown in (b). This example is shown for a noise level of σ/μ = 40%, applied to the diffusivity of the morphogens u and s.

https://doi.org/10.1371/journal.pone.0210088.g005

Numerical evaluation

Numerical forward integration on a 120 × 44 hexagonal lattice with temporal step size 10⁻² was used to evaluate the spatio-temporal evolution of the coupled differential Eqs 1–4. To obtain sufficient data for statistical analysis, the code was parallelized and run on USC’s high-performance supercomputer cluster center. Each computation was assigned to multiple processors at once, running for around 10 hours on each node.

To obtain reliable error bars, at least 50 realizations of each noise level were simulated. The total eye disc size width was chosen such to eliminate edge effects, allowing sufficient space for clusters to form on the edges. Also, a relatively long eye disk length of 120 lattice sites was used is to allow investigation of error propagation of clusters as a function of position from posterior (P) to anterior (A). The simulations were terminated once the morphogenetic furrow had passed throughout all of the eye disc.

Quantitative measures of structural order

While previous quantitative studies have concentrated on characterizing the disorder of the R8 point patterns of experiments and creating minimal underlying models [20, 21], in this work we focus on the robustness of the mathematical model and its structural characteristics in the presence of stochasticity. More specifically, in this study the order of the pattern is quantified when varying the noise of parameters that represent regulatory and other processes, distinguishing from order quantification when changing morphological parameters directly [20, 21]. Curiously, while varying morphological parameters directly, a threshold response is also apparent [21]. Furthermore, the order parameter that is developed and applied is distinct from qualitative measures applied before [7, 8], and is more closely related with the measures introduced in quantitative studies [20, 21]. This analysis will provide insight into pathways with which developmental systems either cope with or succumb to stochasticity beyond a threshold, ultimately leading to malformation.

We wish to analyze the emerging point pattern using appropriate measures of structural order, including translational, bond orientation and variance of nearest neighbor distributions. Details of how these measures are designed and applied are discussed in this section.

Traditional measures for determining structural order have been developed in solid state physics [22–25]. However, in the context of the spatio-temporal formation of patterns in biological developmental models there are additional issues to address. Foremost, the degree of structural order in the final pattern is generally not homogeneous in the presence of stochasticity. While, per initial conditions, the posterior point pattern starts off close to an ordered state, it may result in a strongly disordered anterior region at later stages of development, depending on the degree of stochasticity. This type of disorder is correlated between rows because of the manifestly Markovian mechanism that produces the patterns, i.e. the points on a given row are specified based on the geometry of the points in the previous row [26].

This observation renders traditional solid state measures used to evaluate the structure of homogeneous crystals insufficient [23]. Furthermore, the type of correlated disorder exhibited in the point patterns of the eye disc is unlike random thermal displacements found in atoms or infrequent impurities. Instead, such correlated disorder causes Bragg’s law to become inapplicable, since it is based on the assumption that the underlying unperturbed lattice is periodic [22]. Specifically, the model system is far away from the two instances where a corrected Bragg’s law for imperfect lattices could still be applied. The Debye-Waller approximation [27, 28] applies only for uncorrelated deviations from a perfectly periodic lattice, and the available corrections for correlated deviations [29] apply only in the limit of small deviations and short-ranged Gaussian correlations. In order to study the structure of amorphous solids and general highly disordered materials, as in the case of developmental models, it is therefore necessary to resort to the pair distribution function as a local measure of structural order [23].

The definition of the pair distribution function is given by (7)

In the context of developmental models, this function describes the probability of finding an activated cell at position r, given that another activated cell is located at position r_j. The pair distribution when treated as function of the scalar distance, as in the case of isotropic patterns [23], is denoted by g₂(r) and is called the radial distribution function.

In the following analysis, we will use a variation of the radial distribution function. Practically, in numerical simulations of eye disk development one can calculate the radial pair distribution function only for small distances. Each simulation of the pattern only contains around 40 activated cells. Moreover, the calculation of g₂(r) is limited by the open boundary conditions, because the application of periodic boundary conditions to a single realization and the combination of multiple realizations is unphysical. Therefore, here we focus on a variation of the radial distribution function, that includes only nearest neighbors. The nearest neighbors were identified via the Voronoi tessellation method [30].

Furthermore, we analyze a scalar measure of the spatial order of the patterns, the translational order parameter defined as (8) where η_c is a cutoff limited by the simulation size [24]. This order parameter can be interpreted as the Kolmogorov probability distance [31] between the radial distribution function of the target pattern and the Poisson random point process. (The radial distribution function of a Poisson random point process is the uniform distribution, which in the normalization used in Eq 8 is equal to 1). The translational order parameter, T, is a general order metric used to describe systems independently of the underlying crystal structure. In this setting, we examine how fast the pattern deteriorates compared to the perfect pattern, so we replace the uniform probability distribution with that of the perfect pattern. In addition, since we only considered the radial distribution of nearest neighbors, the cutoff _c in the integration in Eq 8 was replaced with the furthest nearest neighbor distance.

Finally, we study the bond angle order parameter. Contrary to the translational measure, this order parameter evaluates the spatial orientation of vectors connecting the nearest neighbors of all points. It is defined by (9) [25], taking a value of 1 for a perfect hexagonal point pattern and 0 for a completely random pattern. Here N_n is the total number of nearest neighbors in the point pattern, the j’s sum over all activated cells, and the k’s sum over all nearest neighbors of a given reference point. Lastly, θ_jk is the angle of the vector connecting each point with its nearest neighbor with respect to a fixed axis.

The discussion in this section connects and motivates the order measures used in this article with the tools available in the literature. It is important to note that equations Eqs 7 and 8 apply to point patterns in continuous space. In this article, the point patterns are defined by positions of the activated R8 cells which are constrained to lie on a hexagonal lattice. This means that the argument of the pair distribution function becomes discrete. Thus, we are going to consider discrete probability functions. This leads to the scalar pair distribution function to be defined as: (10) where N_bins is the number of bins used, n_j the frequency of activated cells at the jth bin and N is the total number of R8 cells. Similarly, the translational order parameter is defined as: (11)

Probability distribution functions of nearest-neighbor distances and angles.

To quantitatively determine the degree of disorder, a post-processing code is used to record the activated R8 cell positions in the final patterns and generate the probability distributions of nearest-neighbor angles and distances. Firstly, the R8 clusters are refined to single R8 cells by the cluster refinement method explained above. For every R8 cell, the nearest neighbors are identified via the Voronoi tesselation [30].

Next, nearest-neighbor distances and angles of each R8 cell are calculated, and a filter is applied to correct for boundary effects for the nearest neighbor angles. Combining this information for all random realizations, the probability distribution functions of nearest-neighbor distances and angles are determined. Since the cells in the simulation are positioned on a hexagonal lattice, the nearest-neighbor distances and angles take discrete values. As a consequence, the probability distributions are also discrete. They are shown in Fig 6 for the ordered case. These types of probability distribution functions are used to quantify the order of the R8 cell point patterns by calculating the variance and the distance between probability distributions. The same approach can be taken when interpreting experimental data obtained from imaging.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 6. Nearest neighbor (a) distance and (b) angle probability distributions of R8 cell point pattern in the Drosophila eye disc in the absence of stochasticity.

Distances are naturally binned by the available sites on the underlying hexagonal lattice, whereas angular positions are collected in bins of 0.05 radians.

https://doi.org/10.1371/journal.pone.0210088.g006

Even in the zero noise case, the histograms contain multiple peaks in Fig 6. The histogram peaks for the nearest neighbor distances are a consequence of three effects. First, there are two length scales associated with the nearest neighbor distances in the pattern, one for the x-axis and one for the y-axis. Even for a perfect hexagonal lattice there are two lattice constants, one for each dimension. Second, cells in the edges of the simulation grid are placed in a different separation than the cells in the bulk. The two former points result to four peaks. Third, by the definition of nearest neighbor dictated by the Voronoi tessellation, the edge cells along the y-direction boundaries are nearest neighbors of each other creating a fifth peak. The very short peaks that remain in the rightmost side of Fig 6(a) are a consequence of edge effects and the Voronoi tessellation in the x-direction boundary. Finally, slight blurring of each peak is a consequence of clusters forming one or two cells away than their counterparts.

Threshold response

To quantify disorder in the emerging activated R8 cell patterns, the nearest neighbor distance and angle probably distribution functions were computed for various noise levels in the diffusion coefficients D_u, D_s. (Later we will discuss the effect of stochasticity on other model parameters). Their histograms are shown in Fig 7. The binning used for the distance histograms reflects the discreteness of the underlying hexagonal lattice on which the cell centers are placed. For the distance diagrams, all possible nearest neighbor distances on the hexagonal lattice are used as bins. For the angle histograms, since there are many more possibilities, a constant bin size of 0.05 radians was chosen in order to appropriately resolve the distribution.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 7. Nearest neighbor distance (left panels) and angle (right panels) probability distributions generated from the point-pattern in the Drosophila eye disc R8 cell specification for increasing noise levels.

Stochasticity was introduced in the differential equations by drawing the value of the diffusion coefficients D_u and D_s from a normal distribution with mean μ and standard deviation σ. The nearest neighbor distance histograms in (a) to (d) and nearest neighbor angle histograms in (e) to (h) correspond Gaussian stochasticity with standard deviations σ/μ = 0%, 30%, 40%, 60%. While the first two histograms are almost identical, as noise in the diffusivity of the morphogens increases, the peaks get smeared out and the histograms appear wider.

https://doi.org/10.1371/journal.pone.0210088.g007

The angle histograms, unlike the distance histograms, are skewed towards larger angles. This asymmetry makes the analysis of the distributions more complicated, as in this case the variance, paints an incomplete picture. As discussed below, the probability distance order measure eliminates this issue. The angle histograms are skewed as a direct consequence of the cluster refinement process. As seen in the plots of atonal patterns (Fig 4), the onset of disorder is signaled by elongated clusters of activated R8 cells. Since these elongated clusters are brought down to a single cell in the cluster refinement step, and they cast large inhibition radii, the number of nearest neighbors found for the elongated clusters is less than the six found in the perfect pattern. A lesser number of neighbors trivially leads to higher angles in this case. Both histograms, distances and angles, start out sparse for the case without any stochasticity in D_u, with empty bins between peaks, indicative of the highly ordered repeating lattice structure produced by the simulations in this case. As stochasticity increases, the resulting point patterns become aperiodic, leading to denser histograms. Furthermore, the histograms become wider with increasing stochasticity. This is captured in the variance, shown in Fig 8.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 8. Disorder in the R8 cell point pattern of the developing Drosophila eye disc as a function of stochasticity.

Here, the variance of the distributions of (a) nearest neighbor distances and (b) nearest neighbor angles is used to quantify the amount of disorder. As in the previous figure, stochasticity is introduced in the diffusivity of the morphogens D_u and D_s, and the x-axis represents the noise level in terms of the standard deviation of the underlying Gaussian distribution from which this parameter is drawn. As the noise level in the simulation is increased beyond a threshold, the variance grows.

https://doi.org/10.1371/journal.pone.0210088.g008

The variance plots in Fig 8 illustrate the generic nature of robust pattern formation in stochastic developmental models. They show that for small levels of stochasticity (here σ/μ < σ_c/μ = 20%) the ordered pattern remains basically unaffected. However, beyond this threshold the variance starts increasing linearly, both in the angle and distance histograms. This threshold response is a central finding of this study. As discussed below, this is a universal phenomenon, largely independent of which model parameters experience stochasticity and which measures of order are used to assess the system response. It implies that these stochastic models capture biological resilience against stochastic variation up to a certain point, resulting in a regime of deterministic outcomes in spite of stochastic input.

Let us now discuss how this type of threshold behavior is also picked up by the more general measures of order given by Eqs 13 and 14. The results of applying these probability distance measures is shown in Fig 9. Note that saturation observed in the nearest angle distribution variance plot is apparent in both cases, and the responses of both observables (distances and angles) to stochasticity exhibit a universal sigmoidal functional form, implying threshold behavior. These measures reveal the same interesting quality as the variance: the pattern order exhibits an initial resistance to weak input stochasticity that eventually gives way to structural malformation at larger noise levels. The sigmoidal functional form confirms that the mechanism does exhibit robustness for σ < σ_c. The fact that this is observed in the Kolmogorov measure, the fidelity, the bond orientation and variance measures supports the notion that such sigmoidal, threshold response to stochasticity is a universal feature (see supporting information, S4 Fig, for bond orientation sigmoidal response).

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 9. Probability distance measures applied to analyze response to stochasticity in R8 cell point pattern formation.

(a)-(b) are generated using the fidelity, F, and (c)-(d) using the Kolmogorov distance, K, for nearest neighbor distances and angles respectively. Stochasticity was introduced in the model through drawing the parameter D_u and D_s in the differential equations Eqs 1–4 from a normal distribution with varying standard deviation, σ, and mean value, μ. The simulation results were compiled, and nearest R8 cell neighbor angle and distance distributions were obtained. Kolmogorov and fidelity of the resulting distributions where computed with reference to the perfect pattern case. Here, a value of 1 corresponds to a perfectly ordered pattern, and a value of 0 corresponds to a completely irregular pattern. This figure illustrates that the functional form is independent of the measure used.

https://doi.org/10.1371/journal.pone.0210088.g009

Next, we turn to the question of universality with respect to how stochasticity is introduced. We observe that stochasticity in other model parameters also produces threshold behavior, indicating that the observed threshold response is a generic feature of this class of evolutionary models. Specifically, in Fig 10 we show the response to increasing stochasticity levels in the production rate of atonal P_a, the production rate due to the activator S, and the production rate caused by the morphogenetic furrow G. In all these cases, there is also a regime σ < σ_c where the ordered pattern remains intact. For the threshold response in the case of noise in all parameters, except the ones determining h, the two remaining production rates Pu and Ps and for more cases see S2 Fig.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 10. Universal threshold response to stochasticity in different model parameters.

In each sub-plot, stochasticity is introduced in the following parameters of the underlying differential equations: (a) D_u and D_s, (b) P_a, (c) S, and (d) G. Introduction of parametric noise in all of the former parameters, produces a sigmoid response. The order measure used is the fidelity of the nearest neighbor distance probability distributions and sigmoid curves were used to fit the data.

https://doi.org/10.1371/journal.pone.0210088.g010

Next let us address the origin of the observed resilience of this model towards weak noise with σ < σ_c, signified by the initial plateaus observed in Fig 10. The main reason for these wide regimes, exhibiting robustness with respect to parameter stochasticity, is the presence of Scabrous as a diffusible activator [7]. Introducing Scabrous decreases the sensitivity of the patterning mechanism to the shape of the inhibitory regions [7] visualized in Fig 2, and thus to slightly misplaced clusters, making the model more robust with respect to parametric perturbations. This effect can be identified in the ordered state where displaced cluster positions do not lead to dissipation of the order of the pattern, see S1 Fig. While this ingredient is not needed for the pattern to propagate, without this element even low stochasticity levels of D_u cause it to deteriorate, making the model of pattern formation biologically implausible [7]. This effect is quantitatively demonstrated in Fig 11, which compares the response to stochasticity with and without Scabrous as a diffusible activator. In the absence of Scabrous, with the same parameter set and with optimized initial conditions, there is an immediate deterioration of the patterned order upon introduction of infinitesimal noise. In contrast, the presence of Scabrous changes this functional relationship to the observed sigmoid response, thus introducing a resilience scale to the pattern formation mechanism set by the Scabrous production rate P_s and the activation threshold s₁.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 11. Robustness of Drosophila eye disc R8 cell pattern order versus stochasticity, with and without the diffusible activator Scabrous in the model.

The analysis is applied in the same setting as Fig 9, i.e. stochasticity is introduced in the diffusivity of the morphogens D_u and D_s. In the absence of the diffusible activator s, there is a much increased sensitivity to stochasticity for infinitesimal non-vanishing σ. The order measure used is the fidelity of the nearest neighbor distance probability distributions.

https://doi.org/10.1371/journal.pone.0210088.g011

In addition to the robustness introduced by Scabrous, the threshold response of the order of the pattern to stochasticity in the various parameters shown in Fig 10 is a consequence of three further elements: (i) the spatial discreteness of the underlying lattice, (ii) the threshold activated response of cells to morphogen levels, and (iii) propagation of avalanches from anterior to posterior. As we now discuss, the robustness for low stochasticity levels is a consequence of the lattice discreteness and the threshold response of the cells to morphogen concentrations, whereas the sharp decay is a consequence of propagation of avalanches.

To understand how the discreteness of the underlying lattice contributes to robustness for low noise levels, consider the effect of altering the diffusivity of a morphogen. Since the local morphogen concentration drops off at distances on the order of a few cells, and the cells lie on a discrete hexagonal lattice, a small change in the diffusivity radius will not result in any difference unless it leads to the inclusion or exclusion of additional cells. A continuous change in the radius of inhibition will exclude or include new cells when it increases by a factor comparable to the lattice spacing, thus creating a threshold response. Alternatively, consider the effect of the morphogenetic furrow, mathematically denoted by Eq 4, which propagates with a constant speed through the lattice of cells. Two consecutive cells on along the direction of propagation of the morphogenetic furrow will receive the activation signal with a delay lag, δt. Only if the production rates, denoted by G in Eq 1, picked from the normal distributions differ above a finite threshold one will result to a misplaced cell. This effect is shown in the absence of s as a short range activator, when noise in introduced in G, in S5 Fig. An analytical approach to this effect and its contribution in error is carried out in [19].

Second, the threshold response of pattern formation to parameter stochasticity is linked to the threshold activation of cells in response to morphogen levels. The turn-on and turn-off of morphogens in the differential equations is governed by step functions with threshold values, a_a, a_u, u₁, h₁, s₁. This has the consequence that the effect of morphogen concentration is binary, i.e. the parameter variation must be sufficient such that the concentration threshold is crossed for activation or inhibition to occur. Nevertheless, even for more continuous cell responses to morphogen levels, there is always a threshold value and the same argument can be made. To rule out whether the threshold response to stochasticity is an artifact of the binary functional form of the Heaviside functions appearing in Eqs 1–4, further simulations were performed, see S2 Appendix.

Finally, to explain the fast decay of the pattern beyond threshold stochasticity, the propagation of avalanches comes into play, an effect also identified by [7, 8]. Beyond σ_c, R8 clusters start to get significantly elongated and misplaced. The first such elongated R8 cell cluster causes the next row of R8 cell clusters to be misplaced, leading to a cascade onset of pattern disorder. This avalanche effect is quantitatively discussed next, where we analyze the order of the pattern as a function of position.

Increasing disorder of the pattern from posterior to anterior

Here we use the fidelity measure to evaluate error propagation as a function of position on the spatial posterior-to-anterior axis. The mechanism of pattern formation relies delicately on carefully spaced precursor R8 cells that are placed in the posterior region before the growth process described by the differential equations is initiated. Their positions define the subsequent cluster spacing of activated R8 cells for the entire eye disc. In the presence of stochasticity, this information dissipates during the propagation of the morphogenetic furrow from posterior to anterior, since the ommatidial pattern on a column is defined by the shape of the inhibition signals emitted by the ommatidial row before it. It is thus expected that as the morphogenetic furrow travels from posterior to anterior, errors in the pattern order grow in an avalanche fashion. This effect is observed in the simulations, as shown in Fig 12, where the disorder of the pattern clearly increases from posterior to anterior.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 12. Decay of R8 pattern order as a function of position on the eye disc (same parameter set as in Fig 5).

The local order measure is calculated as a function of position, from posterior to anterior. The x-axis refers to regions of the simulated eye disc 0-40, 40-80, 80-120 respectively, whereas the y-axis refers to the fidelity probability distance measure applied to nearest neighbor distance distributions. The three colored lines correspond to various stochasticity levels in D_u and D_s, from σ/μ = 15% to σ/μ = 25% in steps of 5%. Local order generally decreases from posterior to anterior. Once the threshold of σ_c/μ ≈ 20% is crossed, there is a very sharp spatial decay of the order of the pattern, signaling the underlying avalanche effect.

https://doi.org/10.1371/journal.pone.0210088.g012

Within the noise regime studied, the order parameter F decreases by two effects. First, the pattern becomes increasingly disordered as it propagates. Second, a larger part of the pattern becomes disordered. Interestingly, in the high noise regime, the pattern saturates to a value for F that does not increase further from posterior to anterior, see S6 Fig. The higher the noise, the faster it reaches to the saturated value of F. Thus, in the regime of high noise the disorder is largely due to a larger part of the pattern being in the disordered state. Conversely, for low noise regimes, the dominant effect is the intensification of disorder from posterior to anterior. These two effect are also universal, observed for all parameters, since they are the precursors of a threshold response to stochasticity.

Previous studies have not, so far, quantitatively analyzed the consequences of finite system sizes on the order of the pattern. To illustrate these finite size effects on the threshold response quantitatively, in Fig 13 we plot the order parameter obtained from numerical simulations for different eye disc sizes, illustrating the effect of size on the robustness of the system. Here one clearly observes that the larger eye-disc, the sharper the threshold response to stochasticity. Since our numerical simulations were performed on lattices much smaller than the actual eye disc, the observed threshold effect can be extrapolated to be much more pronounced for realistic eye disc sizes.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 13. Pattern order as a function of eye disc size (same parameter sets as in Fig 8).

The threshold response is more pronounced as the length of the eye disc in the direction of the morphogenetic furrow is increased. The order parameter is calculated for cell number 40, 80 and 120 while keeping the size perpendicular to the propagation direction of the morphogenetic furrow at 44 cells. The order measure used is the fidelity of the nearest neighbor distance probability distributions and noise is added on the diffusivity of the morphogens.

https://doi.org/10.1371/journal.pone.0210088.g013

This variation in the degree of dependency of the pattern order measure upon stochasticity levels across different eye disk sizes is analogous to the finite size scaling of order parameters commonly observed in interacting physical many body systems, such as Ising models on lattices. In analogy to phase transitions in the thermodynamic limit of such models, i.e. for infinite system sizes, here we observe precursor sigmoidal response to increasing stochasticity levels that becomes more pronounced for larger system (eye disk) sizes, ultimately culminating into a true phase transition as L_x → ∞. The structure of the underlying system of coupled dynamical equations suggests that this transition belongs to the universality class of directed percolation [32]. While a complete scaling analysis is beyond the scope of this study, it will be provided in a future publication.