Introduction

Development is the process by which the intricate complexity of multicellular organisms is constructed from a single fertilized egg or some simple vegetative structure (Fusco and Minelli, 203). Not many other natural processes lead to so much complexity in such a short time. Development entails a natural process of pattern formation: specific cells and cell types end up in specific positions in space (i.e., anatomy). This process of pattern formation can be seen as the generation of spatial information (i.e., information about where each cell and cell type are) from previous information (e.g., information within the fertilized egg). This latter information includes the DNA but also spatial information in the form of compartments with different proteins and RNAs in different spatial locations within the oocyte. In most species, development cannot proceed if this latter spatial information is eliminated experimentally (Gilbert and Barresi, 2023). This initial spatial information within the oocyte arises from spatial asymmetries in the mother’s gonads or from the environment (Gilbert and Barresi, 2023). Pattern formation in development, thus, does not usually start from a spatially homogeneous initial condition but from an initial condition that is relatively simple but spatially heterogeneous.

Development can be seen as a sequence of transformations between initial developmental patterns and latter developmental patterns (what we call resulting patterns) over developmental time. By developmental pattern, or simply pattern, we mean a specific distributions of cells and gene product concentration over space (Figure 1). In most animals, the earliest patterns arise from the division of the fertilized egg into different cells that, thus, inherit different parts of the fertilized egg and different proteins and RNAs (Gilbert and Barresi, 2023). Some of these latter molecules act as transcriptional factors that lead to the expression of further genes, i.e., the transcription and translation of genes into gene products (Gilbert and Barresi, 2023). Some of these gene products are secreted into the extracellular space. In here, we call these molecules extracellular signals but in the literature they are also called morphogens, growth factors, paracrine factors, etc. (Gilbert and Barresi, 2023).

Figure 1:

Extracellular signals diffuse in the extracellular space and bind to specific receptors in distant cells (Gilbert and Barresi, 2023). Cells can respond to these signals by changing gene expression and, often, by secreting additional extracellular signals and regulating cell behaviors such as cell division, cell contraction, cell adhesion, cell death, etc. (Salazar-Ciudad et al. 2003; Gilbert and Barresi, 2023). The former type of response leads to further changes in gene expression over cells (i.e., further pattern transformations), while the latter leads to cell movement and, consequently, to changes in the distribution of cells and gene expression over space (i.e., further pattern transformations) (Salazar-Ciudad et al. 2003).

How cells respond to extracellular signals depends on the signal receptors and signal transduction pathways they express (Gilbert and Barresi, 2023). Signal transduction pathways are actually networks of molecular interactions that integrate incoming signals to determine cell responses. In here, we use the term gene network to refer to these networks of interactions, even if they also include molecules that are not gene products (Gilbert and Barresi, 2023).

A fundamental question of developmental biology is how pattern formation, or more precisely pattern transformations (Salazar-Ciudad et al. 2003), occur. In this article we restrict ourselves to pattern transformation through gene networks and cell signaling (i.e. no cell division, cell contraction, no extracellular matrix secretion or any cell behaviors leading to cell movement). We ask:

1. Which are the gene networks topologies that can lead to pattern transformation?

2. Can these topologies be classified into a small number of classes with similar topologies and leading to similar pattern transformation?

In this article we only consider non-trivial pattern transformations in which cells do not move. By this we mean pattern transformations in which there are changes on which cells have higher concentrations of some gene product than their immediate neighbors. Non-trivial pattern transformations, thus, imply that at least one gene product exhibits new concentration peaks or changes in the position of concentration peaks (see Figure 1). We exclude the cases in which simply, a gene starts to be express in the same pattern than a gene that was already expressed (see section S1 in SI for a more formal consideration). We only consider developmental patterns that are stable in time.

There are several previous theoretical studies exploring how gene products can be wired to lead to pattern transformations through cell signaling (Salazar-Ciudad et al., 2000, 2001; Cotterel and Sharpe, 2010; Marcon et al., 2016; Zheng et al., 2016; Jimenez et al., 2017; Diego et al., 2018; Leyshon et al., 2021). Some of them are restricted to networks of three gene products (Cotterel and Sharpe, 2010; Zheng et al., 2016) while others restrict themselves to study a specific class of gene network topology (Marcon et al., 2016; Zheng et al., 2016; Jimenez et al., 2017; Rand et al., 2021; Leyshon et al., 2021). In a previous study we addressed the same question that in here but through numerical simulation. We found two classes of pattern-formation gene network topologies (Salazar-Ciudad, 2000). This previous study, however, did not show why the identified topology classes are the only possible ones and, due to its purely exploratory approach, it failed to identify one class. In here we take a more general mathematical approach to identify and characterize all possible classes; with any number of gene products.

It is worth noting that there is an abundant theoretical literature on the topic of gene networks in cell biology (e.g., metabolism, gene regulation for cell basic functions, etc.). However, the topic of this article is fundamentally different to those. Here we are not dealing with single cell gene networks, but with systems of cells with identical gene networks that are coupled through extracellular signaling. Specifically, we are interested in those such networks that lead to the phenomenon of pattern transformation (that is an inherently spatial question).

In this study we consider three broad types of initial patterns (i.e., initial conditions): homogeneous initial patterns, spike initial patterns and combined spike-homogeneous initial patterns. In an homogeneous initial pattern, all gene products have the same non-zero concentration everywhere except for some small random fluctuations due to the noise intrinsic to the molecular level. In a spike initial pattern, the concentration of all gene products is zero everywhere except in a cell. At this cell, one or more gene products have a non-zero concentration. As we will see, the secretion of an extracellular signal from this spike leads to an extracellular concentration gradients centered in the spike. The combined spike-homogeneous initial patterns consist of a spike on an otherwise non-zero homogeneous initial pattern (see Figure 2). Any other initial pattern can be constructed by combining spikes of different heights (i.e., different gene product concentrations) at different positions. We then study which gene networks can transform these initial patterns into other (i.e. resulting patterns) non-trivial ones.

Figure 2:

Methods: The model

Let us consider a set of N_c cells, each with an identical gene network of N_g gene products. For the purpose of clarity we will explain our results as if these cells have a simple arrangement in space (e.g., a 1D line or a 2D square lattice) but, as we will discuss, our results shall apply with the same logic to any distribution of cells in space. Even though we will only use the term gene product, our model should apply to any kind of biological molecule whose concentration changes as a consequence of the concentration of other molecules. We consider two types of gene products: intracellular gene products and extracellular signals. For intracellular gene products, g_i (t, x) denotes the concentration of gene product i inside the cell in position x at time t >0. For extracellular signals, g_i (t, x) denotes the concentration of the extracellular signal i in the extracellular space surrounding the cell in position x at time t >0.

The dynamics of the gene product concentrations in our model obeys the following system of N_g equations (Murray 2002),

where is the vector of concentrations of each gene product at position is a function that takes a vector of concentrations as input and returns another vector of concentrations as output; μ_i is the intrinsic degradation rate of gene product i (i.e., the default rate at which a gene product would be degraded by cellular and extracellular proteases if no other factors affect its degradation); and d_i is the diffusion coefficient of gene product i in the extracellular space.

The last term in (1) is Fick’s second law (Fick, 19855) describing how the concentration of a molecule changes due to diffusion. ∇² denotes the Laplace operator (i.e., in 1D; in 2D, etc.). Hence, our model only considers extracellular signals that diffuse in the extracellular space (i.e., no membrane-bound signals).

The first two terms in system (1) can be understood as reaction terms. Function f describes how each gene product affects the production of any other gene product within cells. f defines a directed graph via its jacobian matrix J (i.e., the matrix of first derivatives of f with respect to each gene product concentration) (see S2 in SI for further details), where each node stands for a given gene product and each edge describes an interaction between gene products (Figure 2B). This directed graph coincides with the gene network and thus, each different J that we consider defines a different gene network.

The parameters of equation (1) and function f are identical in all cells because they are assumed to be encoded in DNA and DNA is the same in all cells of an organism. f does not specify which gene products actually interact in each cell, but which gene products would interact if they happen to be in the same cell at the same time. When and where this would happen depends on the actual dynamics of a gene network on an initial pattern. For the sake of simplicity, we do not explicitly differentiate between transcriptional factors, receptors or other molecules involved in signal transduction: they are just intracellular gene products in a network.

We say that gene product j directly regulates gene product k if the corresponding element of the jacobian matrix is non-zero (i.e., J_kj ≠ 0). This regulation is positive if J_kj >0 (i.e., g _j increases with g_k); or negative if J_kj <0 (i.e., g _j decreases with g_k). We say j is downstream from k if there exists a chain of regulations between gene products going from k to j. Correspondingly, j is said to be upstream from k (see S3 of the SI for a more formal discussion). Notice that a gene product can be both upstream and downstream of a set of gene products, thus forming a regulatory loop.

In contrast to previous studies (Cotterel and Sharpe, 2010; Marcon et al., 2016; Zheng et al., 2016; Jimenez et al., 2017; Diego et al., 2018; Leyshon et al., 2021), our results apply to any gene network with any number of genes and any f as long as the latter fulfills a set of very broad biological and mathematical requirements. (R1) f is continuous and continuously derivable (at least locally). (R2) f is non-lineal. This is because a linear f would only lead to zero or infinity concentrations in the long-term (Kondepudi & Prigogine, 2014; Murray, 2002). (R3) f is a function of g_i but not an explicit function of time or any time derivative of g_i. This means that the regulation of a gene product at a given moment depends only on the concentration of other gene products at that given moment, and not explicitly on past concentrations or their rate of change over time. (R4) f is such that g_i is always non-negative and bounded, that is, gene product concentrations are not negative nor infinite. (R5) f is monotonously increasing or decreasing with respect to each gene product concentration. In other words, the partial derivative of f with respect to each gene product, should have the same sign for any value of g _j,

where f _k is the output of f regarding gene product k (i.e., the k-th component of f), and all gene product concentrations, except forg _j, are kept constant. This final requirement simply states that each gene product interaction is always positive or negative (i.e. the sign of the regulation of k by j does not change with the concentration of j). Although, there are known exceptions to this requirement in biological gene networks (Nelson et al., 2021), this requirement ensures that complex regulation in our model arises from relatively simple interactions between gene products (i.e., gene network topology) rather than from single gene products depending on complex ways on the concentrations of input gene products (i.e., intricate f _k’s). We consider that f can have parameters other than the ones described in (1), as long as these do not change over time and space (i.e., they have to be genetically encoded).

In this article, we assume that our parameter choices for equation (1) and f are such that, in absence of extracellular signals, the intracellular part of gene networks exhibit no complex temporal dynamics such as limit cycles or chaotic behaviors (i.e., their concentrations reach a fixed value). In later sections we discuss some intuitions on how our results may also hold even in the presence of these temporal dynamics.

Finally, we define the topology matrix T of a gene network as the matrix whose elements are − 1, 1 or 0 depending on the sign of the corresponding jacobian element (i.e., T =sgn (J)). Studying gene network topology is important because for most developmental systems we only have information about T, but not so much about J (Gilbert and Barresi, 2023). In the results we show that there are only three classes of gene network topologies leading to non-trivial pattern transformation and discuss which types of resulting patterns can arise from each topological class, and their main properties. However, we only provide necessary topological conditions, that is, we show which resulting patterns can be attained from each topological class, but we say nothing about the specific values of μ_i, d_i and J in (1) for which these patterns actually arise. The results are organized into a set of logical arguments (for the least evident of them, we provide mathematical proofs in the supporting information).

Results

At first sight it may seem that the set of possible gene networks is too vast and the possible dynamics of pattern transformation too complex for any meaningful classifications. In this article we will show that this is not the case.

Without cell movement, the only way cells can change their gene expression over space is through cell communication by extracellular signals. No matter how complex is a gene network, cells in one place need to affect cells in other places for pattern transformation to occur and, without cell movement, this requires cell communication. Any network can be partitioned into a set of subnetworks but given that pattern transformation requires cell communication, we focus on partitioning gene networks into one-signal subnetworks. In this article, a one-signal subnetwork is the set of interactions between the gene products downstream of an extracellular signal (and the signal itself). One-signal subnetworks, or from now on simply subnetworks, can partially overlap since a gene can be downstream of several extracellular signals (see Figure 3).

Figure 3:

Requirements on the intracellular part of gene networks capable of pattern transformation

Here we present two logical requirements that gene network topologies need to fulfill. These are rather trivial but introducing them here facilitates explaining later results.

Requirement I1

For pattern transformation to occur some of the gene products present in the initial pattern must be positively upstream of at least one subnetwork. Without this requirement no signals would be expressed and, thus, no pattern transformations can occur. Similarly, any gene product with a non-zero concentration in the resulting pattern (i.e. the pattern arising from a given pattern transformation) should be positively downstream of at least one gene product present in the initial pattern.

Requirement I2

All gene products are constantly being degraded (second term in (1)). Thus, for a gene product to be present in the resulting pattern it should constantly receive positive regulation from some gene products. This ultimately requires the gene product being within a self-activatory loop or downstream from it (either intracellular or intracellular). This broad case, includes genes that are constitutively expressed, meaning that they are always active regardless of other genes, since this be viewed as these genes being within a self-activatory loop of their own (see section S3 of the SI).

Gene network classification

In this section we explain that all conceivable one-signal subnetworks can be classified into just three classes based on how cells respond to an extracellular signal in terms of secreting, or not, the same extracellular signal (see Figures 3 and S1). Let Abe the most upstream extracellular signal of a given one-signal subnetwork. Then, there are trivially only two options, either Ais not downstream of itself (directly or indirectly), or it is. In the former case, we say that the subnetwork is hierarchical, i.e., class H; while in the latter we say that the subnetwork is emergent. The emergent class can be further divided in the L⁺class, ifAis positively downstream of itself, or class L^-, if it is negatively downstream of itself. In other words, in H subnetworks extracellular signals cannot be within any regulatory loop, while in the L classes they are (either with other extracellular signals or with themselves through intracellular gene products).

Our classification is purely based on topology but, since pattern transformation requires signaling, it is exhaustive: any gene network leading to pattern transformation must include at least one subnetwork, and all one-signal subnetworks fall within these three classes (or their combinations).

Subnetworks can be combined into larger gene networks. We classify whole gene networks in the same way than their composing subnetworks, e.g., a gene network with only H subnetworks is an H gene network. Composite gene networks are labeled according to their composing subnetworks, independently of the number of subnetworks of each class, e.g., a gene network with H and L⁺ subnetworks is an HL⁺ gene network. In the following sections we will explain which classes of gene networks (i.e., combinations of subnetworks) can lead to non-trivial pattern transformations and which ones cannot.

Hierarchical gene networks

Spike initial patterns: Gradient formation in H gene networks

Let us first consider H gene networks consisting of a single H subnetwork with a single extracellular signal. We call these H⁰ networks. From the spike initial pattern, this means that only one cell is secreting extracellular signals (i.e., the one in the spike). Thus, the concentration of extracellular signals in all other cells is fully governed by signal diffusion from this source and its degradation. Indeed, if we consider the set of cells either on the left- or right-hand side of the spike, we can solve equation (1) (see section S5.1 of SI) to show that signal concentrations form an exponential gradient around the spike (see Figure S3),

Each cell at each side of the spike is at a unique distance to the spike and, thus, experiences a unique concentration of A (even if this concentration may barely differ between nearby cells). In that sense, there is a unique relationship between the concentration of Aand the distance to the spike. This correspondence has been used to propose that with a single extracellular signal, any pattern transformation is possible. The only thing that would be required is that cells interpret signal’s concentration in a differentc way for each different resulting pattern attained (Wolpert, 1968, Wolpert, 2016). However, what would that interpretation be, or which would be its underlying gene networks, has not been specified (Horder, 2001). Allegedly, any interpretation should rely on specific gene networks with reasonably simple regulations between gene products, as in our 5 requirement on f. A common idea compatible with this approach would be that cells express specific genes if they receive Aat concentrations beyond some threshold value that is different for different genes. Then different genes may become expressed at different distances to the spike and, thus, a non-trivial pattern transformation would arise (Capek and Müller, 2019; Sharpe, 2019).

There are, however, many other ways in which H⁰ can operate. In the following we derive the minimal general requirements that H⁰ networks should fulfill to lead to non-trivial pattern transformations (see section S5.2 of the SI for details). The first requirement (RH1) is that at least one gene product positively downstream of A, sayk, inhibits another gene product that is also positively downstream of A, say j. In other words, there needs to be some inhibition between gene products and, due to requirement I1, these gene products need to be downstream from A. This requirement is discussed in previous work by other authors (Monteanu et al., 2014), specially for networks with three gene products, but we include it and expanded in here for completeness.

Let us investigate why (RH1) is necessary. If kdoes not inhibit j(i.e., if RH1 does not hold), but Apositively regulates bothkand j, then, according to requirement (R5) onf, bothkand j will increase their concentration when Adoes and vice versa. Over space, this means that jandk would exhibit peaks and valleys of concentration in the same places than A, and thus, no non-trivial pattern transformation can occur. Consequently, any gene product exhibiting a non-trivial pattern transformation has to receive a negative regulation and, since in H⁰ networks there is only one signal, this inhibition should come, directly or indirectly, from A. This fact, together with requirement I1, explains (RH1) (see section S5.2 of SI for details).

For non-trivial pattern formation in H⁰ networks, it is also required that either the activation of k is non-linearly different to the activation of j for different concentrations of A (RH2a) or that the inhibition of jbykis non-linearly higher for high concentrations ofkthan for low concentrations ofk (RH2b) (see Figure S3 and section S5.2 of SI for further details). Let us first explain RH2a (see Monteanu et al., 2014 for a similar discussion). Imagine an H⁰ network fulfilling (RH1), but neither RH2a nor RH2b. In that network, Aregulateskand j in the same way everywhere and, thus, their concentration over space varies in the same way as that of A. This implies that the proportionsg_k / g_Aandg _j/ g_Aare the same all over space and, hence, jandkwill have their peaks of concentration in the same place than A(i.e., no non-trivial pattern transformation is possible). On the contrary, if the network satisfies (RH2a), then the inhibition of kon jis stronger close to the spike (whereg_kis higher because A is higher) than far away from it (whereg_kis lower because A is lower). This way, a valley of j can form close to the spike and, consequently, two peaks of j would form at each side of it (Figure S3). See section 5.2 of the SI for why these interactions have to be non-linear. The same changes in the proportions g_k / g_Aand g _j/ g_Aover space can be attained if the network fulfills requirement (RH2b) (see section S5.2 of SI for formal proofs). Notice that in 2D these networks do not lead to peaks but to rings.

Pattern transformations in H gene networks with several extracellular signals

We call H gene networks to the hierarchical gene networks using more than one extracellular signal. These can also lead to non-trivial pattern transformations (Figure S3). In this case inhibition does not need to occur between at least two genes products downstream of A but can occur between gene products downstream from different extracellular signals secreted from the spike. Then, neither requirement (RH2a) nor (RH2b) are necessary if these different extracellular signals have different d, μ or secretion rates (RH2c). In this latter cases, the different extracellular signals only need to attain different proportions between their concentrations over space (e.g. different decay exponents with distance to the spike due to, for example, different diffusion coefficients; see section S5.3 of SI). If this occurs, then inevitably, some of the extracellular signals will have a higher proportion of their concentration close to the spike than others. As in the previous case then, concentration valleys can form around the spike for those gene products that are inhibited by one signal and activated by the other, directly or indirectly (see Figure S3 and section S5.3 of SI for details).

Pattern transformations in gene products further downstream of extracellular signals

Gene products that acquire a pattern through a H (or H⁰) network can lead to further pattern transformations in downstream gene products without the need for extra extracellular signals. This would occur in those gene networks in which gene products with a pattern activate gene products that are also inhibited by gene products with a different pattern (Figure S4). The pattern of the downstream gene product can then have concentration valleys where its inhibitor gene products have concentration peaks and concentration peaks where its activator gene products have concentration peaks (see section S5.4 of SI for further details). Similarly, a gene product that is positively downstream of gene products with different patterns would also develop a new non-trivial pattern in which concentrations peak would form wherever any of the upstream gene products have concentration peaks (Figure S4). The number of peaks could also be up to the sum of the number of peaks in the upstream gene products (even slightly more as explained in section S5.4 of SI). By hierarchically combining these positive and inhibitory interactions, complex patterns can be produced (Figure S4).

The ensemble of possible pattern transformations from H gene networks and spike initial conditions

For any non-trivial pattern transformation (as long as it is symmetric around the initial spike), there exists a H gene network capable of producing it from a spike initial pattern. To understand that let us first acknowledge that any resulting pattern can be seen as a sequence of concentration peaks of different heights and steepness at different positions. Now, let us consider the so-called diamond network. This simple gene network leads to a two-peak pattern around the initial spike (see Figure 5). By choosing the adequate parameters and f in this network, any combination of height, steepness and positions can be achieved in those peaks (see section S5.5 of SI). This means that by combining different diamond networks (i.e. by having several diamond networks upstream of the same gene product as explained in the previous section), it is possible to develop a pattern with any combination of peak height’s, positions and steepnesses (that is any resulting pattern that is symmetric around the spike) (see section S5.6 in SI).

Pattern formation in H networks from homogeneous initial patterns

No non-trivial pattern transformations are possible in H gene networks from homogeneous initial patterns. In homogeneous initial patterns, the inhibition between gene products that are downstream from the signal would occur in the same way everywhere and, thus, it would not lead to non-trivial patterns (see section S5.7 of SI). The pattern transformations possible from the combined spike-homogeneous initial patterns are similar to those possible from the spike initial conditions (see section S5.1 of SI).

Emergent gene networks

L⁺ gene networks

These are gene networks with one or several L⁺ subnetworks downstream of a gene expressed in the initial pattern (and no other type of subnetwork). From a spike initial pattern, the signal, or signals, in the L⁺ diffuse from the spike and activate their own production in the cells that receive it. As a result, the signal spreads further and ends up being produced by all cells (Figures S8-9). Due to the averaging effect of diffusion and the self-activating nature of L⁺, all cells end up having the same concentration of the signal and, thus, the resulting pattern is homogeneous. The same occurs from homogeneous or spike-homogeneous combined initial patterns.

L^- gene networks

These networks only contain subnetworks with extracellular signals that inhibit their own production, directly or indirectly, in the cells that receive them (i.e., an L^- subnetwork). According to requirement I2, gene networks leading to non-trivial pattern transformations must include one or several self-activatory loops positively upstream of the genes expressed in the resulting pattern. In the case of L^- gene networks, these loops are intracellular (otherwise by our definition of gene network topologies, the network would be L^-L⁺ not L^-). Let l⁺_A denote a positive intracellular loop positively upstream of A.

Pattern transformations from homogeneous and spike initial patterns in L^- subnetworks

Let us first consider the case of an L^-gene network with a single L^- subnetwork and a single l⁺_A that is positively upstream of L^-, but not downstream of it. From a homogeneous initial condition, and because requirement I1, the gene products in l⁺_A will end up being expressed everywhere at the same level regardless of any noise present in the initial pattern since l ⁺_A is a positive intracellular self-activatory loop. Hence, given that A is downstream of l⁺_A, it will also be expressed everywhere at the same level, diffuse and inhibit itself by the same amount everywhere, i.e., no non-trivial pattern transformation (see Figures S8-9).

From a spike initial pattern there will be no non-trivial pattern transformation either. This is because there is no way to spread the expression of the genes in l⁺ away from the spike since this would require an extracellular signal activating gene products in cells away from the spike but, by definition, the only existing extracellular signal inhibits itself. The same occurs if L^- is upstream of l⁺_A but l⁺_A is not upstream of L^- (Figures S8-9).

The only option left for non-trivial pattern transformations is that l⁺_A is both upstream and downstream of A. This way, cells that extracellularly receive A, inhibit its production, while, intracellularly, A promotes its own production via l⁺_A. We call these L^- gene networks, noise-amplifying gene networks (Figure 4), as in (Marcon et al., 2016; Diego et al., 2018).

Figure 4:

Figure 5:

It has been proven that noise-amplifying gene networks can transform homogeneous initial patterns into a specific type of non-trivial patterns (Marcon et al., 2016; Diego et al., 2018). Let us briefly summarize how these arise. Because of molecular noise, the homogeneous initial pattern consists of one or several gene products with random tiny concentration peaks and valleys over an otherwise homogeneous concentration level. Because of requirement I1 and I2, l⁺_A is positively downstream of these gene products and, thus, acquires the same tiny concentration peaks and valleys l⁺_A is a self-activatory loop and, thus, these peaks and valleys start to grow. A diffuses from each growing peak and, by the definition of L^-, it inhibits itself and in the cells receiving it (see Figure S6). Since A activates l⁺_A, inhibiting A means inhibiting l⁺_A too. As a result, the growing peaks of A and l⁺_A inhibit the growth of other peaks of A and l⁺_A nearby. Thus, there is a sort of lateral inhibition between peaks and the resulting pattern consist of concentration peaks (and valleys) that are further a part than in the initial pattern and are much taller (or deeper). The resulting patterns seem random but peaks are more likely to form where the initial pattern had higher tiny random peaks or where there were, by chance, fewer of them. In fact, for each distinct homogeneous initial pattern with random noise, a different resulting pattern can arise (see Figure S6). We call these resulting patterns, amplified noise patterns (as in Marcon 2016).

Combined spike-homogeneous initial patterns

As in the previous initial patterns, only the noise-amplifying gene networks can lead to non-trivial patterns transformations. Noise-amplifying gene networks are known to be able to lead to relatively complex and periodic non-trivial pattern transformations from spike-homogeneous initial conditions (Wang et al., 2022). In this case, the concentration of A starts being higher in the spike than elsewhere. As A diffuses, it leads to the l⁺_A being strongly inhibited in the cells around the spike and thus, to the formation of valleys of concentration for the gene products in l ⁺_A around it (see Figure S6B-C). Because l⁺_A activates A, the cells in these valleys will produce less A and then, cells next to the valley will receive less A. As a consequence there will be less inhibition of l⁺_A in the cells next to the valley and the concentration of both A and l⁺_A will increase in those cells. This increase will, in turn, lead to the formation of a concentration valley further away and these valleys to further peaks and so on. This process continues until all the system is occupied by concentration peaks and valleys (of equal size and shape) (see Figure S6B-C). In 2D there are no peaks but concentric rings of high gene product concentration centered around the spike while in 3D there are concentric spherical shells. We refer to these patterns as ‘frozen-wave’ patterns. As in the case of H networks, these frozen wave patterns are symmetric around the initial spike but, in contrast to H networks, they extend over the whole system.

Changes in the parameters of noise-amplifying gene networks (e.g., in diffusivities), can lead to changes in the number, height or width of concentration peaks and valleys in the resulting pattern but, contrary to what happens for H gene networks, these peaks cannot change independently from each other (see Figure S7). This non-independence is easy to understand from the fact that noise-amplifying gene networks can produce many different concentration peaks from a small number of interactions in the gene network and thus, no specific part of the network is responsible for any specific concentration peak (as it happens for H gene networks). This implies that the ensemble of pattern transformations attainable by these networks are less diverse than the pattern transformations attainable by H gene networks (i.e., the former can only produce patterns with equally sized and shaped peaks, while the latter can produce any pattern that is symmetric around the spike).

Gene networks composed of several L^- subnetworks lead to the same patterns than L^- subnetworks and have very similar properties (see section S6.1 of SI).

Composite gene networks

Subnetworks can be combined in parallel, in series, or both. In parallel means that different subnetworks are upstream of the same set of gene products. Since in this case the gene networks do not regulate each other, it is easy to see that the resulting patterns in the most downstream genes can have concentration peaks and valleys wherever their upstream gene products have them, just as described for H⁰ gene networks above (see Figure S4).

When combined in series, patterned gene products from one subnetwork are upstream of the extracellular signals of other subnetworks. The gene networks composed of L⁺ upstream of H subnetworks are equivalent to H subnetworks acting on a homogeneous initial pattern (since we have seen that L⁺ lead to homogeneous patterns) and, thus, lead to no non-trivial pattern transformation. If H is upstream of L⁺, the attainable pattern transformations are those of H, since L⁺ will homogenize the concentration of its constituent gene products (see section S5.4 of SI for details). More interesting pattern transformations can occur by combining noise-amplifying and H subnetworks acting on combined spike-homogeneous initial patterns in series. Simply, the frozen-wave pattern attainable by noise-amplifying gene network can be locally modified by the H subnetwork so that the peaks close to the initial spike can vary in size (symmetrically around the spike). This can happen both by the H being upstream of the noise-amplifying and the noise-amplifying being upstream of H (see Figure S10).

L⁺L^- gene networks

If an L⁺ subnetwork is upstream of an L^- subnetwork, and not downstream of it, L⁺ is unaffected by L^- and consequently, its genes will be homogeneously expressed regardless of the initial pattern (see Figures S8-9). This reduces to what we have seen for L^- gene networks acting on homogeneous initial patterns (see Figures S8-9). If L^- is upstream of L⁺, and not downstream of it, the genes in L⁺cannot be expressed because the signal in L^-inhibits, by definition, other gene products in the loop (see Figures S8-9).

If L⁺ and L^- are both upstream and downstream of each other, then we have the widely studied Turing mechanisms (Turing, 1952; Murray, 2002; Maini et al., 2006; Meinhardt, 2008). Turing mechanisms include an extracellular signal that promotes its own production (forming an L⁺ subnetwork) and the production of another extracellular signal that inhibits the former (forming an L^- subnetwork). In Turing’s seminal work, each subnetwork was in fact a single molecule (an extracelllarly diffusible one) but it has been widely reported that these mechanisms also apply to gene networks where extracellular signals regulate each other indirectly (though an intracellular part gene network (Salazar-Ciudad, et al. 2000; Satnoianu, et al., 2000; Maini et al., 2006; Meinhardt, 2008; Cotterell & Sharpe, 2010).

Turing gene networks can lead to non-trivial pattern transformation from homogeneous initial patterns (Turing, 1952) and from combined spike-homogeneous initial patterns (Tarumi & Mueller, 1989). The resulting patterns in all cases are very similar, as widely studied over the last 70 years (Turing, 1952; Maini et al., 2006; Meinhardt, 2008). These patterns consist of periodical, regularly spaced peaks and valleys of concentration that repeat themselves over space (Figures 4 and S7). In 1D, these patterns are very similar to those possible by the noise-amplifying gene networks acting on spike-homogeneous initial patterns. In 2D and 3D, however, the resulting pattern consist of spots or stripes (Meinhardt, 1982). In contrast to H gene networks, the spatial information in the initial pattern is not used to construct the resulting pattern and hence, the same, or a very similar, resulting pattern emerges regardless of the initial pattern (Meinhardt, 1982).

As in the case of noise-amplifying gene networks, changes in the parameters of the gene network (e.g., in diffusivities), can change the number, height or width of concentration peaks and valleys in the resulting pattern and, contrary to H gene networks, no peaks or valleys can change their height or width independently from each other (Turing, 1952; Murray, 2002; Maini et al., 2006; Meinhardt, 2008). The reasons for this difference are the same we discussed for noise-amplifying gene networks (see Figure S7). This implies, again, that the ensemble of attainable pattern transformations by Turing gene networks is less diverse than the ensemble of attainable pattern transformations by H gene networks (Salazar-Ciudad, et al. 2001) (Figure 6).

Figure 6:

Naturally, different Turing subnetworks can be combined into composite gene networks. If they are connected in parallel, there can be the same peak addition and subtraction we described for H networks (see Figure S4). By combining Turing subnetworks in series, the resulting pattern of one subnetwork can serve as the initial pattern of the other (Fujita & Kawaguchi, 2013; Moustakas-Verho et al., 2014). These combinations, however, do not lead to different kinds of patterns; the resulting patterns are still the periodic patterns that result from simple Turing networks (possibly combining several wavelengths as in Fujita & Kawaguchi, 2013 and Moustakas-Verho et al., 2014). Thus, combining different Turing gene networks does not lead to qualitatively different resulting patterns nor dynamics, as we show mathematically in section S6.2 of the SI.

Other combinations

In general, the Turing H combined gene networks have variational properties with features of both the Turing and H gene networks (Figure S10). Namely, peaks and valleys will form over the whole system (as in Turing gene networks) and the height of some of these peaks and valley can be varied independently by changes in the parameters of the H subnetwork. These combined properties have been discussed in detail before (Salazar-Ciudad et a., 2001; Miura, 2013; Green & Sharpe, 2015; Glim et al., 2021; Tzika et al., 2023).

From homogeneous initial patterns, combining Turing and noise-amplifying gene networks leads to resulting patterns in between those possible from Turing and noise-amplifying. Essentially periodic patterns with different amplifications of noise will occur (section S6.2 of SI).

Discussion

In this article we have shown that there are only three possible classes of subnetworks (H, L⁺ and L^-) and that these can only be combined in three classes of gene networks (H, Turing and noise-amplifying) to lead to non-trivial pattern transformations. We have also show that gene networks in different classes lead to different types of pattern transformations and have different properties. In section S7 of SI we discuss how these results should hold even if cells have complex temporal internal dynamics (e.g., limit cycles).

The fact that there are only three classes of gene networks leading to pattern transformation can be understood from simple qualitative geometric arguments. From a homogeneous initial pattern there is no way by which gene product concentrations could become different in one spatial part of the system and not on another unless these differences would be ultimately random (as in the noise-amplifying case) or regularly periodic over space (as in in the Turing case). From a spike initial pattern, gene product concentrations can become different in different parts of a pattern because, ultimately, these parts are at different distances from the spike and then exhibit different combinations of concentrations of the signals secreted from it. Based on these spatial differences specific parts of a H gene network can lead to different concentration peaks and valleys and their features (e.g., width, height, etc.).

The fact that there is a limited number of topological classes leading to pattern formation does not depend on the geometry of space (i.e., the distribution of cells in the embryo). The type of pattern transformations for each class does not depend on the geometry of space either, e.g., Turing networks lead to periodic patterns, H networks do not, etc. The geometry of space, however, can change specific aspects of these patterns (Diambra & Costa, 2006; Glimm et al., 2014; Castelino et al., 2020).

All the gene network topologies capable of pattern transformations that we describe have been individually reported before in one way or another (Turing, 1952; Salazar-Ciudad et al., 2001; Cotterell et al., 2015; Wang, 2022). What has not been shown before is that these are the only possible gene network topologies in pattern transformation, given the biologically reasonable restrictions on f that we describe. We think this is important both for theory and practice. For theory it is important because we provide a general description of the possible at the level of pattern transformations and underlying gene network topologies (as long as there is no cell movement). In practice this is also important because it provides a simple guideline for experimental developmental biologist trying to understand pattern transformations in specific organ over a set of developmental stages. Thus, if no cell movements occur during these stages, the underlying gene network should have a topology and variational properties among the ones described in this article. This narrows down significantly the range of possible underlying gene networks and patterning mechanisms to consider when trying to understand the development of a multicellular system. Similarly, this allows to better the study the evolution of gene networks leading to pattern transformations, since then the evolution of the development of any animal species can be understood as the tuning and replacing between these classes of gene networks along the sequence of development (Salazar-Ciudad et al., 2001).

Of the three classes of gene network topologies we identify, two have been widely studied before. Most of the research in Turing gene networks has been theoretical (Turing, 1952; Maini et al., 2006; Meinhardt, 2008) but there is also direct experimental research on the involvement of these gene networks in the development of many organs (e.g., Glover et al., 2017; 2023; Johnson et al., 2023; Tzika et al., 2023; Tseng et al., 2024, just to name some recent ones), although the underlying networks are usually more complex than the ones studied theoretically. Although hierarchical networks are usually not called this way, they constitute the bulk of the gene networks experimentally studied (Salazar-Ciudad et al., 2001) and they are, by far, the easier to understand. This, and the fact that many network seem hierarchical when only some of their interactions are known (Salazar-Ciudad, 2009), may have biased research to focus on them. Although noise-amplifying gene networks seem simpler and easier to understand than Turing networks, they have only been discovered very recently (Cotterell et al., 2015; Marcon et al., 2016; Wang et al., 2022). It is, thus, an open and suggestive question whether this class of network is widely used in pattern transformations in multicellular systems. In fact, it is perfectly possible that organs that are though to use Turing networks may actually be using noise-amplifying networks instead, since the latter produce resulting patterns that are very similar, specially when the organ is effectively 1D.

Acknowledgements

We thank H. Cano-Fernández, S. Green, C. Hedges and A. Loreto-Velázquez for their comments. We are also grateful to the Spanish Ministry of Science and Innovation for funding PID2021-122930NB-100 and CNS2022-135397 to I.S.-C.; and to the Catalan Office of Universities and Research for 2022FI_B 00610 to K.M.-A. This work was likewise supported by the Spanish State Research Agency, through the Severo Ochoa and María de Maeztu Program for Centers and Units of Excellence in R&D (CEX2020-001084-M). We thank CERCA Programme/Generalitat de Catalunya for institutional support.

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Funding

Ministerio de Ciencia e Innovación (PID2021-122930NB-100)

Ministerio de Ciencia e Innovación (PID2021-122930NB-100 and CNS2022-135397)

Ministerio de Ciencia e Innovación (CEX2020-001084-M)

Departament de Recerca i Universitats (2022FI_B 00610)