Encyclopedia of Systems and Control

Living Edition
| Editors: John Baillieul, Tariq Samad

Spatial Description of Biochemical Networks

  • Pablo A. IglesiasEmail author
Living reference work entry
DOI: https://doi.org/10.1007/978-1-4471-5102-9_89-1

Abstract

Many biological behaviors require that biochemical species be distributed spatially throughout the cell or across a number of cells. To explain these situations accurately requires a spatial description of the underlying network. At the continuum level, this is usually done using reaction-diffusion equations. Here we demonstrate how this class of models arises. We also show how the framework is used in two popular models proposed to explain spatial patterns during development.

Keywords

Diffusion Morphogen gradient Pattern formation Reaction-diffusion Turing instability 

Introduction

Cells are complex environments consisting of spatially segregated entities, including the nucleus and various other organelles. Even within these compartments, the concentrations of various biochemical species are not homogeneous, but can vary significantly. The proper localization of proteins and other biochemical species to their respective sites is important for proper cell function. This can be because the spatial distribution of signaling molecules itself confers information, such as when a cell needs to respond to a spatially graded cue to guide its motion (Iglesias and Devreotes 2008) or growth pattern (Lander 2013). Alternatively, information that is obtained in one part of the cell must be transmitted to another part of the cell, as when receptor-ligand binding at the cell surface leads to transcriptional responses in the nucleus. Frequently, describing the action of a biological network accurately requires not only that one account for the chemical interactions between the different components but that the spatial distribution of the signaling molecules also be considered.

Accounting for Spatial Distribution in Models

Mathematical models of biological networks usually assume that reactions take place in well-stirred vessels in which the concentrations of the interacting species are spatially homogeneous and hence need not be accounted for explicitly. These systems also assume that the volume is constant. When the spatial location of molecules in cells is important, the concentration of species changes in both time and space.

Compartmental Models

One way to account for spatial distribution of signaling components is through compartmental models. As the name suggests, in these models the cell is divided into different regions that are segregated by membranes. Within each compartment, the concentration of the network species is assumed to be spatially homogeneous. The membranes in these models can be assumed to be either permeable or impermeable. In permeable membranes, information passes through small openings, such as ion channels or nuclear pores, which allow molecules to move from one side of the membrane to the other. With impermeable membranes, information must be transduced by transmembrane signaling elements, such as cell-surface receptors, that bind to a signaling molecule in one side of the membrane and release a secondary effector on the other side. Note that in this case, the membrane itself acts as a third compartment.

Compartmental models offer simplicity, since the reactions that happen in a single region obey the same reaction kinetics usually assumed in spatially homogeneous models. Even when the reactions involve more than one compartment, as in ligand-receptor binding, this can still be described by the usual reaction dynamics. Care must be taken, however, to account properly for the different effects on the respective concentrations as molecules move from one compartment to another. In models of spatially homogeneous systems, there is little practical difference between writing the ordinary-differential equations in terms of molecule numbers or concentrations, since the two are proportional to each other according to the volume, which is constant. In a compartmental model, if the molecule moves from one compartment to another, there is conservation of molecule numbers, but not concentrations. For example, if a species is found in two compartments with volumes V 1 and V 2 and transfer rates k 12 and k 21 s − 1, then the differential equations describing transport between compartments can be expressed in terms of numbers (n 1 and n 2) as follows:
$$\displaystyle\begin{array}{rcl} \frac{\mathrm{d}n_{1}} {\mathrm{d}t} & =& -k_{12}n_{1} + k_{21}n_{2} \\ \frac{\mathrm{d}n_{2}} {\mathrm{d}t} & =& +k_{12}n_{1} - k_{21}n_{2}. \\ \end{array}$$
Dividing by the respective volumes (\(C_{1} = n_{1}/V _{1}\) and \(C_{2} = n_{2}/V _{2}\)), we obtain equations for the concentrations
$$\displaystyle\begin{array}{rcl} \frac{\mathrm{d}C_{1}} {\mathrm{d}t} & =& -k_{12}C_{1} + k_{21}(\tfrac{V _{2}} {V _{1}} )C_{2} \\ \frac{\mathrm{d}C_{2}} {\mathrm{d}t} & =& +k_{12}(\tfrac{V _{1}} {V _{2}} )C_{1} - k_{21}C_{2}. \\ \end{array}$$
In the former case, the two equations add to zero, indicating that \(n_{1}(t) + n_{2}(t) = \mbox{ constant}\). In the latter, if V 1V 2, then C 1(t) + C 2(t) varies over time as molecules move from one compartment to the other.

Diffusion and Advection

If the distribution of molecules inside any single compartment is spatially heterogeneous, then models must account for this spatial distribution. At the continuum level, this is done using reaction-diffusion equations. The basic assumption is a conservation principle expressed as a continuity equation:
$$\frac{\partial \rho } {\partial t} + \nabla j = f,$$
which relates the changes in the density (ρ) of a conserved quantity (in our case, the concentration of a species: ρ = C) to the flux j and any net production f. In biological networks, the latter represents the net effect of all the reactions that affect the concentration of the species including binding, unbinding, production, degradation, post-translational modifications, etc.
In biological models, the flux term usually comes from one of two sources: diffusion or advection. According to Fick’s law, diffusive flux is proportional to the negative gradient of the concentration of the species as particles move from regions of high concentration to regions of low concentration. The coefficient of proportionality is the diffusion coefficient, D:
$$j_{\mathrm{diff}} = -D\nabla C.$$
Fick’s law describes thermally driven Brownian motion of molecules at the continuum level. If the species is embedded in a moving field, then the flux is proportional to the velocity of the underlying fluid. In this case, we have advective flow:
$$j_{\mathrm{adv}} = vC.$$
In biological systems, advection can arise because of the movement of the cytoplasm, but it can also represent directed transport of molecules, such as the movement of cargo along filaments by processive motors. In general, molecules exhibit both diffusive and advective motion: \(j = j_{\mathrm{diff}} + j_{\mathrm{adv}}\), leading to
$$\frac{\partial C} {\partial t} + \nabla (-D\nabla C + vC) = f,$$
which, under the assumption that the diffusion coefficient and the transport velocity are independent of spatial location, leads to the reaction-diffusion-advection equation:
$$\frac{\partial C} {\partial t} = D\nabla ^{2}C - v\nabla C + f.$$
Being a second-order partial differential, the solution requires an initial condition and two boundary conditions. Common choices for the latter include periodic (e.g., in models of closed boundaries) or no-flux (to describe the impermeability of membranes) assumptions.

Measuring Diffusion Coefficients

Invariably, solving the reaction-diffusion equation requires knowledge of the diffusion coefficient of the molecule. Experimentally, this can be done in a number of ways. In fluorescence recovery after photobleaching (FRAP), a laser is used to photobleach normally fluorescent molecules in a specific area of the cell. As these “dark” molecules are replaced by fluorescent molecules from non-bleached areas, the fluorescent intensity of the bleached area recovers. Higher diffusion leads to faster recovery. The time to half recovery, τ 1 ∕ 2, can be used to estimate D. If recovery occurs by lateral diffusion, then
$$D = \frac{r_{0}^{2}\gamma } {4\tau _{1/2}}$$
where r 0 is the 1 ∕ e 2 radius of the Gaussian profile laser beam and γ is a parameter that depends on the extent of photobleaching, which ranges from 1 to 1.2 (Chen et al. 2006).
These days, it is increasingly common to measure lateral diffusion coefficients by observing the trajectory of single molecules. A molecule with diffusion coefficient D undergoing Brownian motion in a two-dimensional environment is expected to have mean-square displacement (MSD) equal to
$$\langle r^{2}\rangle = 4Dt.$$
Thus, the coefficient D can be obtained by measuring how the MSD changes as a function of the time interval t. This method can also show if the molecule is undergoing advection in which case
$$\langle r^{2}\rangle = 4Dt + v^{2}t^{2}.$$
This super-diffusive behavior can be seen in the concave nature of the plot of \(\langle r^{2}\rangle\) against t. This plot will also reveal barriers to diffusion. For example, if the molecule is confined to move in a circular region of radius a, then, as t increases, \(\langle r^{2}\rangle\) cannot exceed a 2.
Both these methods work best for molecules diffusing on a membrane. For molecules diffusing in the cytoplasm, the three-dimensional imaging required is considerably more difficult, particularly since the diffusion of particles in the cytoplasm (D ∼ 1–10 μm2 s − 1) is usually orders of magnitude greater than for membrane-bound proteins (D ∼ 0. 01–0. 1 μm2 s − 1). In this case, an analytical expression can be used to estimate the diffusion coefficient. The diffusion coefficient of a spherical particle of radius r moving in a low Reynolds number liquid with viscosity η is given by the Stokes-Einstein equation:
$$D = \frac{k_{B}T} {6\pi \eta r} .$$
The exact viscosity of the cell is unknown, but estimates that η is approximately five times that of water lead to diffusion coefficients of cytoplasmic proteins that match those measured using FRAP.

Diffusion-Limited Reaction Rates

Even in compartments that are considered well stirred, the diffusion of molecules is necessary for reactions to take place. In particular, before two molecules can react, they must come together. To see how diffusion influences this, suppose that spherical molecules of species A and B with radii r A and r B , respectively, come together to form a complex AB at a rate k d . This rate represents the likelihood that molecules of A and B collide at random and hence will depend on the diffusion properties of the two species. The molecules in this complex can dissociate at rate k d or can be converted to species C at rate k r . Thus, the overall reaction involves two steps:
$$\displaystyle\begin{array}{rcl} & A + B{ k_{d} \atop { \rightleftharpoons \atop k_{d}^{{\prime}}} } AB& \\ & AB\stackrel{\rightarrow }{k_{r}}C. & \\ \end{array}$$
Assuming that the system is at quasi-steady-state, that is, the concentration of AB is constant, the effective rate of production C is given by
$$k_{\mathrm{eff}} = \frac{k_{d}k_{r}} {k_{d}^{{\prime}} + k_{r}}.$$
There are two regions of operation. If k d  ≫ k r , then \(k_{\mathrm{eff}} \approx k_{r}(k_{d}/k_{d}^{{\prime}})\). In this case production is said to be reaction limited. If \(k_{d}^{{\prime}}\ll k_{r}\), then \(k_{\mathrm{eff}} \approx k_{d}\) and production is diffusion limited. In this case, it is possible to find k d as a function of the species’ diffusion coefficients.
Assume that species A is stationary, in which case the effective diffusion is the sum of the two diffusion coefficients: \(D = D_{A} + D_{B}\). The concentration of species B depends on the distance away from molecules of A. Because we assume that the reaction rate is fast, at the point of contact (\(r^{\star } = r_{A} + r_{B}\)) the concentration is zero since any molecules of AB are quickly converted to C. At the other extreme, as r → , the concentration approaches the bulk concentration B 0. According to Fick’s law, this concentration gradient causes a flux density given by \(j = -D(\partial B/\partial r)\). The total flux into a sphere of radius r is then
$$J = 4\pi r^{2}j = -4\pi Dr^{2}\frac{\partial B} {\partial r} ,$$
which, at steady state, is constant. Solving this equation for B(r) using the two boundary equations leads to a flux
$$J = -4\pi DB_{0}r^{\star },$$
from which we have that
$$k_{d} = 4\pi Dr^{\star }.$$
A typical value for k d , using the Einstein-Stokes formula, is
$$4\pi \left (2 \times \frac{k_{B}T} {6\pi \eta (r^{\star }/2)}\right )r^{\star } = \frac{8k_{B}T} {3\eta } \approx 10^{3}\,\upmu \mathrm{m}^{-1}\,\mathrm{s}^{-1}.$$

Spatial Patterns

The effect of spatial heterogeneities has been of long interest to developmental biologists, who study how spatial patterns arise. Two distinct models have been proposed to explain how this patterning can arise. Here we introduce these models and discuss their relative merits. Though usually seen as competing models, there is recent evidence suggesting that both models may play complementary roles during development (Reth et al. 2012).

Morphogen Gradients

A morphogen is a diffusible molecule that is produced or secreted at one end of an organism. Diffusion away from the localized source forms a concentration gradient along the spatial dimension. Morphogens are used to control gene expression of cells lying along this spatial domain. Thus, a morphogen gradient gives rise to spatially dependent expression profiles that can account for spatial developmental patterns (Rogers and Schier 2011).

The mathematics behind the formation of a morphogen gradient are relatively straightforward. The concentration of the morphogen is denoted by C(x, t). There is a constant flux (j 0) at one end (x = 0) of a finite one-dimensional domain of length L, but the morphogen cannot exit at the other end. The species diffuses inside the domain and also decays at a rate proportional to its concentration (\(f = -kC\)). Thus, the concentration is governed by the reaction-diffusion equation:
$$\frac{\partial C} {\partial t} = D\frac{\partial ^{2}C} {\partial x^{2}} - kC,$$
with boundary conditions: \(D\frac{\partial C} {\partial x} = -j_{0}\) at x = 0, and \(D\frac{\partial C} {\partial x} = 0\) at x = L. We focus on the steady state:
$$\frac{\mathrm{d}^{2}\bar{C}} {\mathrm{d}x^{2}} = \frac{k} {D}\bar{C},$$
so that the initial condition is not important. In this case, the distribution of the species is given by
$$\bar{C}(x) = \frac{\lambda j_{0}} {D} \frac{\cosh ([L - x]/\lambda )} {\sinh (L/\lambda )} .$$
Thus, the shape of the gradient is roughly exponential with parameter \(\lambda = \sqrt{D/k}\), known as the dispersion, which specifies the average distance that molecules diffuse into the domain before they are degraded or inactivated. Equally important in determining the gradient, however, is the spatial dimension (L) relative to the dispersion, \(\Phi = L/\lambda\), a ratio known as the Thiele modulus. If \(\Phi \ll 1\), then the concentration will be approximately homogeneous. Alternatively, \(\Phi \gg 1\) leads to a sharp transition close to the boundary where there is flux and a relatively flat concentration thereafter.

Though morphogen gradients are commonly used to describe signaling during development, where the gradient can extend across a number of cells, the mathematics described above are equally suitable for describing concentration gradients of intracellular proteins. In this case, the dimension of the cell has a significant effect on the shape of the gradient (Meyers et al. 2006).

As discussed above, morphogen gradients are established in an open-loop mode. As such, the actual concentration experienced at a point downstream of the source of the morphogen will vary depending on a number of parameters, including the flux j 0 and the rate of degradation k. Moreover, because the concentration of the morphogen decreases as the distance from the source grows, the relative stochastic fluctuations will increase. How to manage this uncertainty is an active area of research (Rogers and Schier 2011; Lander 2013).

Diffusion-Driven Instabilities

In 1952, Alan Turing proposed a model of how patterns could arise in biological systems (Turing 1952). His interest was in explaining how an embryo, initially spherical, could give rise to a highly asymmetric organism. He posited that the breaking of symmetry could be a result of the change in the stability of the homogeneous state of the network which would amplify small fluctuations inherent in the initial symmetry. Turing sought to explain how these instabilities could arise using only reaction-diffusion systems.

To illustrate how diffusion-driven instabilities can arise, we work with a single two-species linear reaction network:
$$\frac{\partial } {\partial t}\left [\begin{array}{*{10}c} C_{1} \\ C_{2} \end{array} \right ] = A\left [\begin{array}{*{10}c} C_{1} \\ C_{2} \end{array} \right ]+ \frac{\partial ^{2}} {\partial x^{2}}D\left [\begin{array}{*{10}c} C_{1} \\ C_{2} \end{array} \right ]$$
where \(A = \left [\begin{matrix}\scriptstyle a_{11}&\scriptstyle a_{12} \\ \scriptstyle a_{21}&\scriptstyle a_{22}\end{matrix}\right ]\) specifies the reaction terms and the diagonal matrix \(D = \left [\begin{matrix}\scriptstyle D_{1}&\scriptstyle 0 \\ \scriptstyle 0 &\scriptstyle D_{2}\end{matrix}\right ]\) the diffusion coefficients.
We assume that, in the absence of diffusion, the system is stable, so that \(\det (A) > 0\) and \(\mathrm{trace}(A) < 0\). When considering diffusion in a one-dimensional environment of length L, we must consider the spatial modes, which are of the form exp(iqx). In this case, stability of the system requires that \(\mathrm{trace}(A - q^{2}D) < 0\) and det(A − q 2 D) > 0. The former is always true, since \(\mathrm{trace}(A - q^{2}D) = \mathrm{trace}(A) - q^{2}(D_{1} + D_{2}) < \mathrm{trace}(A) < 0\). However, the condition on the determinant can fail since
$$\det (A - q^{2}D) = D_{ 1}D_{2}q^{4} - q^{2}(a_{ 22}D_{1} + a_{11}D_{2}) +\det (A).$$
(1)
Since det(A) > 0, diffusion-driven instabilities can only occur if the term \(a_{22}D_{1} + a_{11}D_{2} > 0\), by which it follows that at least one of a 11 or a 22 must be positive. Since \(\mathrm{trace}A < 0\), it follows that the diagonal terms must have opposite sign. Usually, it is assumed that a 11 > 0 and that a 22 < 0. Since det(A) > 0, it follows that a 12 and a 21 must also have opposite sign.

These requirements in the sign pattern of the two molecules lead to one of two classes of systems. In the first class, known as activator/inhibitor systems, the activator (assume species 1) is autocatalytic (a 11 > 0) and also stimulates the inhibitor (a 21 > 0), which negatively regulates the activator (a 12 < 0). In the other class, known as substrate-depletion systems, a product (species 1) is autocatalytic (a 11 > 0), but in its production consumes (a 21 < 0) the substrate (species 2) whose presence is needed for formation of the product (a 12 > 0). Note that both systems involve an autocatalytic positive feedback loop (a 11 > 0), as well as a negative feedback loop involving both species (a 12 a 21 < 0).

The stability condition also imposes a necessary condition on the dispersion of the two species, (\(\lambda _{i} = \sqrt{D_{i } /\vert a_{ii } \vert }\)), since
$$a_{22}D_{1} + a_{11}D_{2} > 0\qquad \Longrightarrow\qquad -\lambda _{1}^{2} +\lambda _{ 2}^{2} > 0$$
Thus, the species providing the negative feedback (inhibitor or substrate) must have higher dispersion (λ 2 >  λ 1). This requirement is usually referred to as local activation and long-range inhibition.

These conditions are necessary, but not sufficient. They ensure that the parabola defined by Eq. 1 has real roots. However, when diffusion takes place in finite domains, the parameter q can only take discrete values \(q = 2\pi n/L\) for integers n. Thus, for a spatial mode to be unstable, it must be that det(A − q 2 D) < 0 at specific values of q corresponding to integers n. If the dimension of the domain is changing, as would be expected in a growing domain, the parameter q 2 will decrease over time suggesting that higher modes may lose stability. Thus, the nature of the pattern may evolve over time.

Over the years, Turing’s framework has been a popular model among theoretical biologists and has been used to explain countless patterns seen in biological systems. It has not had the same level of acceptance among biologists, likely because of the difficulty of mapping a complex biological system involving numerous interacting species into the simple nature of the theoretical model (Kondo and Miura 2010).

Summary and Future Directions

Spatial aspects of biochemical signaling are increasingly playing a role in the study of cellular signaling systems. Part of this interest is the desire to explain spatial patterns seen in subcellular localizations observed through live cell imaging using fluorescently tagged proteins. The ever-increasing computational power available for simulations is also facilitating this progress. Specially built spatial simulation software, such as the Virtual Cell, is freely available and tailor-made for biological simulations enabling simulation of spatially varying reaction networks in cells of varying size and shape (Cowan et al. 2012).

Of course, cell shapes are not static, but evolve in large part due to the effect of the underlying biochemical system. This requires simulation environments that solve reaction-diffusion systems in changing morphologies. This has received considerable interest in modeling cell motility (Holmes and Edelstein-Keshet 2013).

Another aspect of spatial models that is only now being addressed is the role of mechanics in driving spatially dependent models. For example, it has recently been shown that the interaction between biochemistry and biomechanics can itself drive Turing-like instabilities (Goehring and Grill 2013).

Finally, we note that our discussion of spatially heterogeneous signaling has been based on continuum models. As with spatially invariant systems, this approach is only valid if the number of molecules is sufficiently large that the stochastic nature of the chemical reactions can be ignored. In fact, spatial heterogeneities may lead to localized spots requiring a stochastic approach, even though the molecule numbers are such that a continuum approach would be acceptable if the cell were spatially homogeneous. The analysis of stochastic interactions in these systems is still much in its infancy and is likely to be an increasingly important area of research (Mahmutovic et al. 2012).

Cross-References

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© Springer-Verlag London 2014

Authors and Affiliations

  1. 1.Electrical & Computer Engineering, The Johns Hopkins UniversityBaltimore, MDUSA