1.1 Tissue growth regulation

The proportions of the human body, including the size and relative proportions of different organs, are primarily determined by a combination of genetic factors [1] and biological signaling pathways during growth, development and regeneration [2,3]. Genes play a critical role in determining how organs develop and maintain proper proportions relative to one another. For example, genes regulate the production of growth factors, hormones, and proteins that influence cell growth, organ formation, and the overall shape of the body. Hox genes, in particular, are a set of genes that determine the basic body plan and help control the spatial arrangement of organs and tissues during embryonic development. They ensure that organs form in the correct places and proportions.

Growth factors like fibroblast growth factor (FGF), transforming growth factor (TGF), and insulin-like growth factor (IGF) regulate cell division and growth [4,5]. They ensure that different tissues and organs grow at appropriate rates and stop growing when they reach a certain size. Hormones such as growth hormone (produced by the pituitary gland) and thyroid hormone play critical roles in controlling growth in specific organs and overall body development [6]. For example, during puberty, growth hormone and sex hormones cause rapid growth in bones and muscles. Different organs have intrinsic growth programs that regulate their size independently to a certain degree. For example, the heart develops and grows at a rate that is proportional to the overall body size to ensure proper circulation [7]. The liver has a remarkable ability to regenerate and maintain its proportional size even if a portion is removed [8]. The body uses feedback mechanisms to regulate growth. As certain tissues grow, signals are sent to stop further growth when a critical size is reached. This prevents organs from growing too large or too small in relation to the rest of the body.

Although genetics plays an important role, environmental factors such as nutrition, physical activity, and exposure to toxins can influence how the body grows and how the organs maintain their proportion [2]. For instance, poor nutrition during childhood can lead to stunted growth and smaller organ size.

In general, a complex interaction between genetic, hormonal, and environmental factors governs the proportionality between different organs of the human body. Considering genetic and environmental factors as given, in this work we will focus on feedback mechanisms and tissue cross-talk based on the exchange of signaling molecules such as hormones and growth factors.

1.2 Tissue cross-talk

Tissue cross-talk refers to the communication and interaction between different tissues within an organism, particularly through signaling molecules like hormones, cytokines, growth factors, and extracellular vesicles. This communication is crucial for coordinating tissue growth, repair, and maintenance in a multicellular organism. During the regulation of tissue growth, these interactions help ensure that different tissues grow in harmony and adapt to changing physiological demands [2].

There are different mechanisms of tissue cross-talk in growth regulation. In the case of paracrine signaling, cells release signaling molecules (such as growth factors) that act on nearby cells. For instance, fibroblasts in connective tissue release growth factors like fibroblast growth factor (FGF), which promote the proliferation of epithelial cells in the skin or other tissues [9]. In endocrine signaling, hormones are released into the bloodstream and affect distant tissues. An example is growth hormone (GH) produced by the pituitary gland, which stimulates growth in bones and muscles [10]. Insulin-like growth factors (IGFs), produced in response to GH, also play a key role in tissue growth regulation [11]. Extracellular vesicles such as exosomes can carry proteins, lipids, and RNA between tissues, allowing them to communicate over long distances. These vesicles can regulate cell proliferation, differentiation, and tissue repair by transferring growth-promoting or inhibitory signals [12].

The immune system is also involved in tissue growth regulation. For example, macrophages and other immune cells release cytokines that promote or inhibit cell proliferation and tissue growth, depending on the context (e.g., during injury repair or in chronic inflammation) [13].

Among key examples of tissue cross-talk in growth regulation we can cite bone-muscle cross-talk [14]. During skeletal growth, there is coordination between bone and muscle tissues. Muscle-derived growth factors, such as myokines, influence bone development, while bone-derived factors, such as osteokines, regulate muscle growth. This interaction is crucial for proper musculoskeletal development and maintaining function throughout life.

Adipose tissue (fat) and muscle also engage in cross-talk through hormones like leptin and adiponectin [15]. Leptin, secreted by fat cells, influences energy metabolism and muscle function, while muscle-derived myokines regulate fat metabolism. This interaction is important in obesity, where dysregulation can lead to abnormal tissue growth and metabolic issues.

In cancer, the cross-talk between tumor cells and surrounding stromal tissue is a key aspect of tumor growth regulation [16]. Stromal cells can secrete growth factors and remodel the ECM to support tumor expansion, while tumor cells can influence the surrounding microenvironment to suppress immune responses or promote angiogenesis.

Tissue cross-talk plays a dynamic role in coordinating the growth and adaptation of different tissues, allowing the organism to maintain a balanced physiological state. When these interactions are disrupted, it can lead to abnormal growth patterns or diseases such as cancer, fibrosis, or tissue degeneration.

1.3 Models of endocrine system and tissue growth control

Endocrine axes govern and regulate the secretion of hormones from different glands through a sequence of signals. They maintain homeostasis, regulate a plethora of physiological processes (e.g. growth and development, reproductive functions), correlate responses of organisms to environmental changes (e.g. adaptation to stress). Endocrine axes rely on negative feedback loops to maintain appropriate balance between levels of different target hormones thus regulating diverse physiological phenomena. We will discuss the biological mechanisms of such feedbacks in Sect 5.

Mathematical models provide indispensable tools to study tissue cross-talk and related time oscillations in neuroendocrinology [17], energy homeostasis [18], stress response [19], and other body systems [20]. One of the first ODE model of growth processes based on negative feedbacks was presented in [21]. This approach is commonly used in more recent works (see, e.g., [22]). Gradient scaling models and temporal dynamics models in the regulation of drosophila wing disk are reviewed in [2,23,24]. Mathematical models of tissue regeneration are presented in [25].

The main difference and the novelty of the present work is that we consider growing tissue as a spatially distributed system. This approach is quite common in tumor growth models, wound healing or morphogenesis (see, e.g., [26]) but there are few spatial models of tissue growth regulation with systemic feedback. We can cite the models of infection progression with a negative feedback by the adaptive immune response [27–29]. From the point of view of modelling and analysis, the interest of such models is that they contain integral terms and represent nonlocal reaction-diffusion equations with some different properties in comparison with the classical reaction-diffusion models.

In the next section, we will develop mathematical models of tissue growth regulation with systemic feedback. We will make abstraction of specific tissues and signaling molecules in order to develop a general framework of such models. Specific tissues and their cross-talk will be studied in the other works. Sect 3 is devoted to differentiation and growth of a single tissue, and Sect 4 to coordinated growth of two tissues. In Sect 5, we discuss different mechanism of tissue growth control in relation to the modelling results, and conclude this paper in the last section.

2.1 Endocrine regulation of a single tissue

The cell concentration u(x,t) in the tissue is described by the equation

(1)

where C is the concentration of some biochemical substance (growth factor, hormone) regulating tissue growth. We take into account random cell motion described by the diffusion term. The function F(u,C) is considered in the form

(2)

The first term in this function describes cell proliferation rate and the second term cell death. The logistic term u(1–u), commonly used to model tissue growth and cell proliferation, captures the phenomenon of density-dependent proliferation, where cell division slows down as cell density increases. This is a well-established concept both biologically [30–32] and mathematically [33–35].

The extended proliferation term in tissue growth models introduces density-dependent signaling effects—particularly autocrine and paracrine signaling—which modulate cell division beyond the basic logistic growth. The factor reflects positive feedback where cells stimulate their own proliferation or that of their neighbors through chemical signaling. As such, many cells produce growth factors (e.g., EGF, FGF, TGF-β) that stimulate their own division (autocrine) or that of nearby cells (paracrine). The local concentration of these factors increases with cell density u, hence justifying a term like to represent enhanced division [36–38]. In models of tumor growth or wound healing, the proliferation rate is often modeled as a function of local signal concentration, which in turn depends on u [39]. Models explicitly accounting for autocrine loops introduce proliferation terms with overlinear growth to reflect increasing signaling strength with density [40,41].

The functions f₀(C) and g₀(C) show the dependence of cell proliferation and death on endocrine signaling. Endocrine signaling profoundly influences both cell proliferation and death across tissues by modulating signaling pathways, transcription factors, and feedback loops. This regulation is not only experimentally established but also supported by a range of mathematical models, which help in predicting dynamics under normal and pathological conditions. Hormones produced by endocrine glands travel through the bloodstream and affect target tissues by binding to specific receptors, regulating gene expression, and thereby influencing cell proliferation. As such, estrogens stimulate the proliferation of breast epithelial cells. Estrogen binds to estrogen receptors (ER), leading to activation of genes promoting cell cycle progression [42]. Insulin and IGF-1 promote proliferation in various tissues, including muscle, liver, and even in tumor cells. Insulin-like growth factor 1 (IGF-1) activates PI3K/AKT and MAPK pathways, which promote survival and proliferation [43]. Endocrine factors also regulate programmed cell death, which is crucial for tissue homeostasis. In particular, cortisol (glucocorticoid) induces apoptosis in immune cells such as lymphocytes [44]. This is crucial during inflammation resolution. Thyroid hormones regulate apoptosis during brain development and metamorphosis in amphibians [45]. Endocrine regulation is considered in mathematical models of cancer [46,47], hematopoiesis [48], immune response [49].

Endocrine signaling is characterized by hormones secreted into the bloodstream by endocrine glands (e.g., pituitary, thyroid, adrenal glands) that regulate the activity of distant target tissues. However, the regulation is not unidirectional. There is substantial evidence that target tissues themselves produce feedback signals, often in proportion to their mass, activity, or functional state, to modulate the upstream endocrine output. This allows the organism to match hormonal stimulation to physiological need. Among many examples of such feedback regulation between the target tissue and endocrine signaling, we can cite hypothalamic-pituitary-endocrine axis [50,51], erythropoiesis regulation [52,53], insulin and glucose homeostasis [54,55], bone mass and FGF23 [56,57], muscle-liver feedback [58,59].

Thus, target tissues often secrete feedback molecules that serve as informative cues about their volume, activity, or functional state. Denote by B its level in the organism. Then

(3)

Its production rate is proportional to the total tissue volume,

(for the space variable x considered on the whole axis). The second term in the right-hand side of Eq (3) describes its degradation or depletion.

This substance B is transported to another organ or tissue by blood flow and stimulates there production of the endocrine signaling molecule C regulating tissue growth in Eq (1):

(4)

Here C is its concentration in the organism (or its level in blood). It acts on the cells of the tissue where its depletion is proportional to the total cell concentration J(u) (Fig 1). Note that the characteristic time scale of hormone distribution by the blood circulation is essentially less than the characteristic time of cell division and tissue growth. Therefore, hormone redistribution is considered as instantaneous, and the process of this redistribution is not considered in the models of tissue growth and regeneration.

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Fig 1. Schematic representation of the model.

Growing tissue produces some signaling molecule B with the production rate proportional to the total cell concentration . It is transported by the blood flow to the controlling organ and stimulates there production of a feedback signaling molecule C which can amplify or inhibit tissue growth.

https://doi.org/10.1371/journal.pone.0319120.g001

Problem (1)-(4) represents a combination of a reaction-diffusion equation for the variable u(x,t) as a function of space and time with ordinary differential equations for B(t) and C(t). Eq (1) will be considered either on the whole axis or in a bounded interval with Neumann boundary conditions. Note that all variables in these equations are dimensionless. As such, u(x,t) is the cell concentration, that is the number of cells in unit volume, normalized to its maximal value. Dimensionless equation (1) is obtained from the dimensional equation by division on the maximal cell concentration. Parameter a in this equation has a meaning of cell proliferation rate normalized by the maximal cell concentration, that is, relative increase of cell concentration in a unit volume during a unit time. Parameter b characterizes the influence of cell-cell interaction on their division rate. Diffusion coefficient D has dimension , where s_un is a space unit, t_un is a time unit. Functions f and g represent dimensionless rate coefficients. Coefficients characterize the rates of production or degradation of signaling molecules B and C.

Since production and redistribution of B and C can be considered as fast compared to the cell division and death, we can use a quasi-stationary approximation in Eqs (3), (4) setting zero the time derivatives:

Therefore,

can be substituted into Eq (1).

Substituting all these expressions in Eq (1) we obtain the following equation

(5)

All coefficients in this equation are some positive constants. We consider it either on the whole axis or on a bounded interval with no-flux (Neumann) boundary conditions. We will specify the functions f(C) and g(C) below.

2.2 Tissue cross-talk and coordinated growth

Tissue cross-talk refers to the biochemical communication between different tissues via signaling molecules such as cytokines, growth factors, hormones, or extracellular vesicles. This communication is crucial for maintaining homeostasis and coordinating complex physiological processes like development, regeneration, and immune responses. The coordinated growth of tissues via cross-talk ensures that organs develop proportionately and functionally integrate during embryogenesis, wound healing, and disease.

Tissues secrete signaling molecules that act on nearby (paracrine) or distant (endocrine) cells to modulate their behavior, including proliferation and apoptosis. As such, in limb development, mesenchymal and epithelial tissues communicate via Fibroblast Growth Factors (FGFs) and Sonic Hedgehog (Shh). FGFs from the apical ectodermal ridge stimulate mesenchymal proliferation, while Shh from the zone of polarizing activity influences patterning and growth coordination [60].

Tissue cross-talk during organogenesis involves reciprocal signaling loops, ensuring proportional and coordinated organ development. For example, in the liver–pancreas axis, FGF and BMP signals from the cardiac mesoderm and septum transversum mesenchyme guide hepatic specification, and later liver-derived signals modulate pancreatic islet development [61]. In metabolic regulation of adipose–muscle tissue cross-talk, adipose tissue secretes adipokines (e.g., leptin, adiponectin) and muscle secretes myokines (e.g., IL-6, irisin), which regulate each other’s growth and function [58].

In tumors, the tumor microenvironment (fibroblasts, immune cells) and cancer cells engage in cross-talk that regulates tumor cell proliferation and apoptosis. Though pathological, this illustrates the broader principle of cross-tissue signaling influencing growth. In particular, cancer-associated fibroblasts (CAFs) secrete TGF-β, VEGF, and other growth factors that stimulate tumor growth and angiogenesis [62].

In bone marrow–immune system interactions, hematopoietic stem cells (HSCs) in bone marrow receive signals from bone-forming osteoblasts and vice versa. Osteoblasts secrete osteopontin, which affects HSC quiescence and proliferation. This interaction ensures that the expansion of the bone matrix and the replenishment of blood cells are coordinated [63].

Thus, tissue cross-talk via diffusible factors, extracellular vesicles, or direct cell–cell contact ensures synchronized growth by adjusting proliferation/apoptosis rates based on systemic needs, mediating feedback loops during development and regeneration, responding to physiological and pathological stimuli.

We now develop a mathematical model for the description of coordinated growth of two tissues. Consider two tissues with normalized concentrations of cells u(x,t) and v(x,t), respectively. These concentrations are described by the following system of equations:

(6)

(7)

Each of these equations is similar to Eqs (1), (2) described in Sect 2.1. As before, the right-hand side of Eq (6) describes random cell motion, cell proliferation and death. Cell proliferation rate is proportional to their concentration u and to the logistic term (1–u) describing density-dependent proliferation. The factor shows that proliferation rate increases with u due to local cell-cell communication. Finally, the factor shows how the proliferation rate depends on the cytokines produced by both tissues. The properties of this and other functions will be specified below. Cell death rate also depends on the concentrations of cytokines through the function . A similar equation is considered for the concentration v.

According to the biological data on tissue cross-talk described above, each tissue produces a signaling molecule acting on the rates of cell proliferation and death. Their concentrations are described by the equations:

(8)

(9)

Here

are the total tissue volumes (for the problem considered on the whole axis). In order to simplify the model, we consider that the same molecule c₁ (c₂) produced by the first (second) tissue participates in the regulation of both tissues. In a more general case, these signaling molecules can be different.

There are different cases according to the form of the functions :

• Cell death is promoted by cells of the same type and down-regulated by cells of the different type, cell proliferation is independent of them,(10)
• Cell proliferation is promoted by cells of the different type and down-regulated by cells of the same type, cell death is independent of them,(11)
• Cell death and proliferation can depend on signaling with c₁ and c₂.

Summarizing these different conditions, we assume that each tissue down-regulates its own growth through a negative feedback determined by the tissue volume. On the other hand, each tissue promotes growth of the other one (Fig 2). These assumptions are confirmed by some biological observations [64]. Moreover, according to this work, cessation of developmental growth is, in the final stages, due more to an increase in the rate of cell loss than to a reduction in the rate of cell proliferation.

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Fig 2. Schematic representation of the model for the coordinated growth of two tissues.

Each of them down-regulates its own growth and up-regulates growth of the other one.

https://doi.org/10.1371/journal.pone.0319120.g002

3.1 Bifurcation of spatially distributed solutions

For simplicity of calculations, we suppose that . Let g(c) be a non-negative continuous function defined for . The particular case is studied in [65].

Consider Eq (1) on a bounded interval [0,L] with the Neumann boundary conditions:

(2)

For a positive homogeneous in space stationary solution of problem (1), (2), , and it can be found as a solution of the equation

(3)

Suppose that such solution exists and denote it by u₀.

We linearize Eq (1) about this solution and obtain the eigenvalue problem:

(4)

or, taking into account (3),

(5)

with the boundary conditions

(6)

We consider the eigenfunctions

and determine the corresponding eigenvalues:

(7)

The properties of the eigenvalues are formulated in the following theorem.

Theorem 3.1. Suppose that

(8)

and

Then is a positive eigenvalue with the maximal real part,

The assertion of the theorem follows directly from (7). It means that the loss of stability of the homogeneous in space solution occurs with a space-dependent eigenfunction (see, e.g., [66], p. 528). Therefore, this instability leads to the bifurcation of a space-dependent solution. This is different for the equation without the integral term. In this case, , and spatial structures do not emerge (see more details in [65]).

Note that if b>a and g(u) = kg₀(u), where g₀(u) is a positive growing function such that g₀(0) = 0, then the conditions of the theorem are satisfied for all k sufficiently large and all D sufficiently small.

Example. Let and a = 0. Then the instability conditions can be written in the following form [65]: , . Note that the instability emerges for the values of L in some bounded interval. This is different in comparison with the Turing instability for which L should only exceed some minimal value.

Some examples of solutions bifurcating from the spatially uniform solution are shown in Fig 3. Let us note that for an asymmetric initial condition, solution converges to the half-pulse solution, while for a symmetric initial condition to the whole pulse solution. Since the initial condition is symmetric, it is orthogonal to the first eigenfunction, and the solution remains in this subspace. Since the second eigenvalue is also positive for these values of parameters, the solution converges to a symmetric pulse solution.

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Fig 3. Numerical simulations of equation (1) with and , , .

Left: D = 50, the initial data are: 0.1H(x–250–50)H(250–x). Right: D = 20, the initial data are: .

https://doi.org/10.1371/journal.pone.0319120.g003

Taking into account different possible applications and wide variation of biological parameters, the choice of parameter values is motivated by the purposes of modelling, such as the conditions of the emergence of spatial structures and different regimes of tissue growth.

3.2 Existence and stability of pulses on the whole axis

We consider the stationary equation

(9)

on the whole axis. We look for a positive solution of this equation vanishing at infinity,

(10)

Consider the auxiliary problem

(11)

and denote by w_h(x) its positive solution. Then solution of the equation

(12)

provides a solution of problem (9), (10).

Problem (11) has a solution if b > a > 0 and a<h<h_*, where h_*>a is a positive number which can be determined analytically (see p. 4 in [65]). Moreover,

We can now formulate the existence result.

Theorem 3.2. Suppose that b > a > 0, or . Then problem (9), (10) has a positive solution.

Consider some examples.

Examples. If g(c) = kcⁿ with some positive k and n, then the conditions of the theorem are satisfied. They are also satisfied for negative n, but these two cases are different. Introduce the function and consider the equation . If it has a single solution h₀, then for n > 0 and for n < 0. This is related to stability and bifurcations of solutions. If n = 0, then Eq (9) is independent of J(u) and the pulse is unstable [67].

Remark. If the function f(J) is not constant, then equation

(13)

can be reduced to an equation similar to (9) by the introduction of a new function .

3.3 Numerical simulations

Some examples of numerical simulations are shown in Figs 4 and 5. If conditions of Theorem 2.2 are satisfied, then there exists a stationary pulse solution. For the values of parameters in Figs 4 (left), it is stable, and the solution of Eq (1) converges to it. On the contrary, for the values of parameters in Figs 4 (right), the pulse does not exist, and we observe wave propagation.

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Fig 4. Numerical simulations of Eq (1) with and , .

Left: solution converges to a stationary pulse, . Right: solution propagates as a wave, .

https://doi.org/10.1371/journal.pone.0319120.g004

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Fig 5. Numerical simulations of Eq (1) with and , , .

The left panel shows convergence of solution to the stationary pulse, . The corresponding function J(u)(t) is shown in the right panel.

https://doi.org/10.1371/journal.pone.0319120.g005

Under the conditions of the theorem, if solution h of Eq (12) is sufficiently close to the value h_*, then the pulse solution exists, it is wide and top-flat (Fig 5, left). The figure shows convergence of solution of the initial boundary value problem to the stationary pulse solution. Let us note that the initial condition is small. Contrary to the conventional bistable reaction-diffusion equation, here the solution can grow even for any small initial condition. In fact, the presence of the integral in the equation changes its type from the monostable case for small J(u) to the bistable case for large values. Thus, solution grows, takes the form of a flat pulse, then decreases its height and increases its width. The right panel in this figure shows the convergence of J(u)(t) to its limiting value.

The decrease of the plateau value in Fig 5 is determined by the density-dependent cell proliferation. If the decrease of the proliferation rate is faster, then the decrease of the plateau value is not so essential. For example, if we impose the condition that cells stop proliferation if they touch each other and proliferate otherwise, then for a uniform cell distribution, proliferation rate becomes , where for and . An example of such simulation is presented in Fig 6.

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Fig 6. Numerical simulations of Eq (1) with and , , .

The left panel shows convergence of solution to the stationary pulse, . The corresponding function J(u)(t) is shown in the right panel. The proliferation rate is given by the function with the density dependence approximating a step-wise constant function .

https://doi.org/10.1371/journal.pone.0319120.g006

4.1 Existence of pulses

We consider quasi-stationary approximations in Eqs (8), (9). Then

(1)

and Eqs (6), (7) become as follows:

(2)

(3)

where

We look for a positive stationary solution w(x),z(x) of this system of equations on the whole axis:

(4)

(5)

vanishing at infinity:

(6)

Consider the auxiliary problem

(7)

(8)

(9)

and denote its solution by (w_p(x), z_q(x)). This solution provides a solution of problem (4)-(6) if

(10)

and

(11)

Lemma 4.1. The function is positive and continuous in the interval for some , for p = a₁, and as . The function is positive and continuous in the interval for some , for q = a₂, and as .

Proof. By definition, w_p(x) is a positive solution of Eq (7) vanishing at infinity. It is independent of q. The function can have from one to three non-negative zeros. Suppose that there are three of them. This is the case if

Denote by w₀(h) the maximal solution of the equation F_h(w) = 0. It is known that Eq (7) has a positive solution w_h(x) vanishing at infinity if and only if . This inequality holds for .

Denote . Then . If , then , uniformly in x on every bounded interval and, consequently, (see [65] for more detail).

Next, suppose that and set . Then Eq (7) can be written as follows:

(12)

Let us introduce a new function u(y) by the equality . Then it satisfies the equation

(13)

where prime denotes the derivative with respect to y. If b > a and ε is sufficiently small, then this equation has a positive solution vanishing at infinity. Then

as .

The second part of the lemma for can be proved similarly. The lemma is proved.

Lemma 4.2. Suppose that system (10), (11) has a solution for , in some parameter range. Then if and only if .

Proof. Suppose that but remains bounded. Then the left-hand side of equality (10) tends to infinity, while the right-hand side remain bounded since . The second case is proved similarly. The lemma is proved.

Lemma 4.3. If

(14)

then for p and q satisfying (10), (11), convergence does not hold.

Proof. If this convergence occurs, then . From Eqs (10), (11) we obtain

Multiplying these equations, we get

The expression in the left-hand side converges to . This contradiction proves the lemma.

We begin the analysis of the existence of solutions of system of (10), (11) with the case . Then, multiplying these equations, we obtain , where . The function is defined on the interval or . Together with the function , they are defined on the intersection of the intervals

We recall that , , and these functions are positive inside their intervals of definition.

Assume that and consider the function

Since and , then H(p₁) = 0. Next, equalities and implies that . Hence, equation (equivalent to Eq (10)) has a solution in this interval. We proved the following result.

Proposition 4.4. Suppose that and

(15)

Then problem (4)-(6) has a positive solution. If the inequality is opposite, such solution does not exist.

We will now consider the case without the assumption that some coefficients vanish.

Theorem 4.5. If and

(16)

then problem (4)-(6) has a positive solution.

Proof. If , then Eq (10) with respect to p has a solution in the interval for any fixed. Indeed, for p = a₁, , , and

On the other hand, as , and the last inequality becomes opposite for p sufficiently close to p_*. Therefore, Eq (10) has a solution for some intermediate value of p. Denote it by S(q). Then, , as .

Similarly, If , then Eq (11) with respect to q has a solution in the interval for any . We denote it by T(p), and , as .

We need to verify that the curve and intersect for some , . This point of intersection provides solution of system (10), (11) and, consequently, of problem (4)-(6). We will show that condition (16) implies that the curve is above in some vicinity of the point (Fig 7). Then the curves intersect inside the rectangle.

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Fig 7. Schematic representation of the curves and on the (p,q)-plane.

The former starts at the lower boundary of the domain and tends to the point . The latter starts at the left boundary and tends to the same point. If the first curve is above the second curve near this point, as shown in the theorem, they have a point of intersection inside the domain.

https://doi.org/10.1371/journal.pone.0319120.g007

We consider the inverse function to the function . For simplicity of notation, set

Then Eqs (10), (11) become as follows:

We obtain from these equations:

Multiplying the first equation by T(p) and equating the right-hand sides, we get the equality:

(17)

where

Since , then by virtue of condition (16),

Furthermore, , where M is some positive constant independent of p.

Eq (17) implies that for p sufficiently close to p_*. Indeed, if for some sequence p_i such that , then from (17),

Since the first term in the right-hand side of this inequality is larger than 1, and the second one converges to 0, we obtain a contradiction.

Thus, we proved that

(18)

If is a monotone function of q for q sufficiently close to q_*, then we conclude that R(p)>T(p).

Monotonicity of is not needed if we want to verify that for some p₀. This is sufficient for the intersection of the curves and for the existence of solution. We take a value q₀ of for which if q<q₀. Existence of such q₀ follows from the fact that tends to infinity. Then for any q₁ such that it follows that . We choose p₀ such that . It follows from (18) that . Then .

4.2 Existence of waves

System of Eqs (2), (3) cannot have travelling wave solutions with one of the limits at infinity different from zero since the integrals J₁(u), are not defined in this case. Instead of travelling waves in the classical definition, we will consider the solution of the Cauchy problem converging to two waves moving in the opposite directions, one to minus infinity, another one to plus infinity. In this case, the integrals are well defined. Let us proceed to the description of such solutions and to the conditions of their existence.

Consider the equations

(19)

(20)

where P and Q are some positive constants. If

(21)

then these are bistable equations. In this case, each of the equations

has three non-negative solutions, and . Eqs (19), (20) have travelling wave solutions with the limits [0,u₂] and . Moreover, there are some values and (the same as in Lemma 4.1) such that the inequalities

(22)

provide the positivity of the wave speeds. Denote the wave speeds by c₁ and c₂, respectively. If 0<P<a₁, then the corresponding equation is monostable, the wave still exists, and we denote by c₁ the minimal wave speed. Similarly, if 0<Q<a₂, c₂ is the minimal wave speed for the second equation.

Consider the solution u(x,t) of the Cauchy problem for Eq (19) with a sufficiently large initial condition u(x,0) vanishing at infinity, as . Then this solution approaches two waves propagating in the opposite directions and as uniformly on every bounded interval. Therefore, as . Similarly for Eq (20), as . We call such solutions expanding wave solutions. We have

System (2), (3) can be reduced asymptotically for large time to system (19), (20) if the following relations are satisfied:

(23)

We take into account here that c₁ and u₂ depend on P, c₂ and depend on Q,

Multiplying these two equalities, we obtain

Since , then

On the other hand, P is defined in the interval (0,p_*). These two intervals intersect if

(24)

It is the same condition as in Proposition 4.4 for the existence of pulses. Assuming that it is satisfied, we consider the equation

(25)

with respect to P, where

Note that . Moreover, if P = p₀, then , and . If P = p_*, then Q<q_*, and c(Q)>0. Therefore , F(p_*) = 0. Moreover, functions u₂(P) and are bounded from above and from below by some positive constants. Therefore, Eq (25) has a solution. We proved the following theorem.

Theorem 4.6 If system of Eqs (2), (3) has an expanding wave solution, then inequality (24) is satisfied.

This assertion does not give sufficient conditions of the existence of such solutions. However, for constant P and Q such solutions do exist. Their existence can be proved for time-dependent P and Q converging to some limits. So, we can expect existence of such solutions for the coupled problem (2), (3).

Conditions of Theorem 4.5 exclude the existence of such solutions, while conditions of Proposition 4.4 admit them. Combining these results with the results of numerical simulations discussed below, we can conclude that waves with positive speeds (Theorem 4.6) exist together with unstable pulses (provided by Proposition 4.4), but not with stable pulses. This is similar for the scalar equation and monotone systems for which unstable pulses exist if and only if the wave speed is positive [68]. Stable pulses do not exist for the scalar equations and monotone systems [67].

The results on tissue growth can be summarized in terms of the non-dimensional parameter

characterizing the strength of the feedback. If S>1, then the solutions converge to a stable pulse. If the inequality is opposite, they propagate as a wave.

4.3 Numerical simulations

According to the analytical results presented above, inequality

(1)

is associated with the existence of expanding solutions, that is, waves with positive speeds. Theorem 4.6 proves that it is a necessary condition of the existence of such solutions. Numerical simulations confirm their existence (Fig 8). Note that the wave speeds for the two components of solutions are, in general, different from each other.

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Fig 8. Time profiles of the function (left) and of the function (right) for , , , .

The initial condition is a piece-wise constant function; . For these values of parameters and . Dynamics of solutions corresponds to expanding waves.

https://doi.org/10.1371/journal.pone.0319120.g008

If inequality (1) is opposite and conditions of Theorem 3.5 are satisfied, then there exist stationary pulses (Fig 9). We note that the components u₁ and u₂ of solution of problem (4)-(6) are coupled only through the values of their integrals. Therefore, for any solution u₁(x), u₂(x) of this problem, then functions u₁(x), also satisfy it for any real h. This property is illustrated in Fig 9 (left). Increase of the value of k₁ decreases the component u₁ of the solution (Fig 9, right). It also leads to the decrease of u₂ through the integral J(u₁).

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Fig 9. Left panel: profiles of the pulses and for , , , .

The initial data for u₁ and u₂ are given by rectangles shifted by 30 space units. Right panel: profiles of the u₁ pulses for , , and fixed n₁ = 2. The maximal value of the profiles decreases with k₁ equal respectively to 0.75, 1, 2, 3. Other parameters: , , and .

https://doi.org/10.1371/journal.pone.0319120.g009

Let us now consider the case (cf. Proposition 4.4). If condition (1) is satisfied, then numerical simulations show that the stationary pulses are unstable, and we observe wave propagation (Fig 10). If inequality (1) is opposite, waves with negative speeds are observed (Fig 11). In agreement with Proposition 4.4, there are no stationary pulses.

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Fig 10. Time profiles of the function (left) and of the function (right) for , , , . Other parameters: .

The initial conditions are given by piece-wise constant functions, .

https://doi.org/10.1371/journal.pone.0319120.g010

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Fig 11. Time profiles of the function (left) and of the function (right) for , , .

The initial condition is a piece-wise constant function. At time t = 581 ⋅ 100 the functions u₁(x,t) and u₂(x,t) are point-wise smaller than 10⁻⁵: the solution converges to zero. The values of parameters: , , and . In agreement with Proposition 4.4, no pulse solution is observed.

https://doi.org/10.1371/journal.pone.0319120.g011

Convergence of solution to a wide flat pulse is shown in Fig 12. In the beginning of simulation, solution resembles a travelling wave. After some time, it slows down and stops. Biologically, this example illustrates coordinated growth of two tissues to their final sizes. We use this example for numerical simulations of tissue regeneration. We cut the first tissue, while the second tissue is not changed. In the model, this means that we consider the initial condition with a partially truncated first component, while the initial condition for the second component represents the same stationary pulse as in the previous simulation. The solution of this problem converges to the same pulse solution. The integrals of the first and second components of the solution are shown in Fig 13. The integral of the first component monotonically converges to its stationary value, while the integral of the second component decreases in the beginning and grows later. This decrease in the beginning of simulations shows that the second tissue decreases its size to adjust to the first tissue.

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Fig 12. Time profiles of the function (left) and of the function (right) for , , .

The initial condition is a piece-wise constant function. Solution converges to the stable pulse. Note that the x-scales in the two figures are different. The values of parameters: , , and .

https://doi.org/10.1371/journal.pone.0319120.g012

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Fig 13. Integrals (left) and (right) in numerical simulations of tissue regeneration.

For the same values of parameters as in Fig 12, the initial condition u₁(x,0) corresponds to the truncated stationary pulse, and for u₂(x,0) to the complete pulse. The solution converges to the same stationary pulse. The integral of the first component of the solution monotonically increases, while for the second component, it first decreases, then grows to its stationary value.

https://doi.org/10.1371/journal.pone.0319120.g013

5.1 Chalones, growth-inhibitory feedback mechanisms, functional feedbacks

Chalones are tissue-specific, secreted factors that inhibit further cell proliferation in the tissue of origin. The concept of chalones originated with the hypothesis that organ size is regulated by locally produced substances that act to limit growth when a critical mass is achieved [64,69]. Chalones are secreted in proportion to the size or cell number of the tissue. As the tissue grows, their concentration increases locally or systemically, inhibiting further proliferation.

Chalones are critical for maintaining tissue homeostasis and preventing excessive growth. The disruption of chalone signaling can result in unregulated tissue growth, contributing to tumorigenesis. For example, loss of myostatin expression in some cancers correlates with tumor progression and metastasis.

The concept of chalones is being discussed during already more than a century with some examples and counter-examples (see the reviews in [2,64,70]). In the modern biological literature, it is accepted that it can work for some organs (see Table 1 in [69]), but it is not universal. Other mechanisms are also involved in tissue growth regulation depending on the tissue in question and physiological conditions (embryogenesis, development, trauma, neoplasia).

Growth is often regulated through negative feedback loops, which maintain a balance between proliferation and tissue size. These feedback loops can involve systemic signals (e.g., hormones) or local factors (e.g., chalones, mechanical stress).

Local feedback mechanisms include contact inhibition, where high cell density suppresses cell division, is a classic feedback mechanism. Mechanical cues, such as stiffness of the extracellular matrix (ECM), can signal cells to reduce proliferation as tissue tension increases. Molecules like FGF (fibroblast growth factors) and VEGF (vascular endothelial growth factor) operate in local feedback loops to regulate organ development and maintain proportions.

Parabiosis studies in mice (joining circulatory systems of two animals) have shown that systemic factors can rejuvenate aged tissues, but intrinsic local feedback mechanisms primarily govern growth control.

Another growth control is based on functional feedback mechanisms that use organ performance or output to regulate growth. Unlike chalones, which depend on mass or cell number, functional feedback ensures that tissue size matches physiological needs.

Examples of functional control include liver and bile acid flux. Bile acids synthesized by the liver are recirculated through the enterohepatic system. When bile acid flux increases due to enhanced digestion demands, hepatocyte proliferation is induced to expand liver size. Conversely, reduced bile acid flux signals that the liver has reached a sufficient size.

Another example concerns kidney and functional compensation. Following unilateral nephrectomy, the remaining kidney undergoes hypertrophy to compensate for lost function. The signal for this compensation may involve increased serum creatinine levels or local mechanical stress. Importantly, this compensatory growth primarily involves hypertrophy rather than hyperplasia. In thyroid and endocrine glands, negative feedback loops involving hormone levels (e.g., TSH and thyroid hormone) regulate the size and activity of endocrine organs. Hypothyroidism leads to compensatory thyroid hypertrophy (goiter), while hyperthyroidism suppresses TSH secretion, reducing thyroid size.

In distinction from chalones, functional feedback is performance-driven, whereas chalone mechanisms are based on physical size or cell number. Functional feedback often operates through systemic signaling, as seen in liver regeneration involving bile acids or kidney compensation mediated by metabolic demands.

5.2 Models of growth with negative feedbacks

The mechanisms of tissue growth control presented above act through negative feedback. Biological mechanisms of this feedback and their implementations in the models can be different. A common feature of these mechanisms, including functional feedback, is that there is some characterization of the tissue (signal, function) proportional to its size J(u). If we denote its level by B, then we obtain Eq (3) for its time evolution.

From this point on, these mechanisms begin to differentiate. Global (systemic) feedback can be expressed by another signaling molecule C produced by some other organ and described by Eq (4). This endocrine signaling can up-regulate cell death in Eq (1) or down-regulate its proliferation.

Functional feedback, which is also systemic, is determined by some given value B₀ required by the organism. Therefore, tissue growth is proportional to (B₀−B) or, in the dimensionless form, to (1–J(u)) entering as a factor in the tissue proliferation rate. From the modelling point of view, this is similar to systemic feedback down-regulating cell proliferation in the previous paragraph.

Local (autocrine, paracrine) feedback, associated with chalones (though their action can also be systemic), decreases cell proliferation. However, the quantity of substance B with respect to a single tissue cell, that is remains constant since . Therefore, its action does not depend on the tissue volume, and cannot control its growth.

There is a principal difference between the local feedback, where the total quantity of substance B is proportional to J(u), and systemic feedback, where the concentration (level) of C in blood (or in the whole organism) is proportional to J(u). In the first case, as discussed above, the quantity of B with respect to a single cell remains approximately constant, and feedback intensity does not depend on the tissue volume (Fig 14). But relative concentration of B can increase with time and act as growth inhibitor or growth promoter that decreases expression of their growth [69]. In the second case, depletion of C in the tissue is negligible compared to the whole organism, we do not divide C by J(u). In this case, feedback intensity depends on the tissue volume J(u).

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Fig 14. Local (left) and systemic (right) feedback on tissue growth by some factors produced by the tissue.

In the local case, increasing the tissue twice (), increases the total amount of the produced factor. However, its amount with respect to the unit tissue volume remains the same. Therefore, local feedback cannot determine the tissue size. In systemic feedback, instead of the total amount of the produced factor in the whole organism, we measure its level (concentration). Increasing twice the tissue, we also increase twice this level. Its feedback on the unit tissue volume also increases.

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Thus, we model systemic feedback, the mechanism of which can be related to chalones and to other negative feedbacks, or to the functional feedback.

5.3 Modelling results and biological interpretations