^{1}

^{2}

^{1}

^{3}

^{4}

^{*}

Conceived and designed the experiments: JMG ZN BNK. Performed the experiments: JMG ZN BNK. Analyzed the data: JMG ZN BNK. Contributed reagents/materials/analysis tools: JMG ZN BNK. Wrote the paper: JMG ZN BNK.

The authors have declared that no competing interests exist.

The temporal and stationary behavior of protein modification cascades has been extensively studied, yet little is known about the spatial aspects of signal propagation. We have previously shown that the spatial separation of opposing enzymes, such as a kinase and a phosphatase, creates signaling activity gradients. Here we show under what conditions signals stall in the space or robustly propagate through spatially distributed signaling cascades. Robust signal propagation results in activity gradients with long plateaus, which abruptly decay at successive spatial locations. We derive an approximate analytical solution that relates the maximal amplitude and propagation length of each activation profile with the cascade level, protein diffusivity, and the ratio of the opposing enzyme activities. The control of the spatial signal propagation appears to be very different from the control of transient temporal responses for spatially homogenous cascades. For spatially distributed cascades where activating and deactivating enzymes operate far from saturation, the ratio of the opposing enzyme activities is shown to be a key parameter controlling signal propagation. The signaling gradients characteristic for robust signal propagation exemplify a pattern formation mechanism that generates precise spatial guidance for multiple cellular processes and conveys information about the cell size to the nucleus.

Living cells detect environmental cues and propagate signals into the cell interior employing signaling cascades of protein modification cycles. A cycle consists of a pair of opposing enzymes controlling the activation and deactivation of a protein, where the active form transmits the signal to the next cascade level. A crucial challenge in cell and developmental biology is to understand how these cascades convey signals over large distances and how spatial information is encoded in these signals. With the advent of advanced imaging techniques, there has been emerging interest in understanding signal propagation in cells and tissues. Based on a simple cascade model, we determine the conditions for signal propagation and show how propagating signals generate spatial patterns that can provide positional information for various cellular processes.

Cascades of covalent protein modification cycles convey signals from cell-surface receptors to target genes in the nucleus. Each cycle consists of two or more interconvertible protein forms, for example, a phosphorylated and unphosphorylated protein, and an active, phosphorylated protein signals down the cascade. In eukaryotes, post-translational protein modifications include phosphorylation of Tyr, Thr and Ser residues, ubiquitylation, acetylation or sumoylation of Lys, methylation of Arg and Lys, and other modifications

While signaling cascades were studied experimentally and theoretically for more than half a century, most studies disregarded the spatial aspects of signal propagation, considering one or more well-mixed compartment(s) with no variation in spatial dimensions. The stationary and temporal behavior of protein modification cascades was extensively analyzed, starting from pioneering numerical simulations by Stadtman and Chock

External signals received at the plasma membrane have to propagate across the cell to reach their targets, and, therefore, protein diffusion and active transport can change quantitative and qualitative aspects of output signaling by protein cascades

Precipitous gradients of phosphorylated kinases can impede information transfer from the plasma membrane to distant cellular locations, such as the nucleus. In the Ras/Raf/Mek/ERK (MAPK) cascade, Ras, Raf and, partially, MEK activation is localized to the plasma membrane, whereas MEK and ERK deactivation by phosphatases occurs in the cytoplasm. Calculations

Spatial separation of opposing enzymes, such as kinase and phosphatase, are hallmarks of protein modification cascades, including MAPK cascades. Here we consider a cascade of protein modification cycles, where each cycle consists of inactive and active forms of a signaling protein, and the active form catalyzes the activation of the protein at the next level down the cascade (

A simplified model, which neglects protein sequestration effects

Spherical symmetry simplifies analysis of signaling in three dimensions, as the protein concentrations become functions of the radial distance and time only

When diffusivities (

Assuming that the kinases and phosphatases follow Michaelis-Menten kinetics, at each level the phosphatase rate depends on the phosphorylated form concentration, ^{−1}).

It is convenient to use the normalized protein concentrations

In this section we study the case when the total concentrations are small compared to the Michaelis constants, or in terms of the non-dimensional parameters, ^{2} s^{−1},

In addition to solving Eq. (5) numerically, we will explore analytically how the kinase activation profiles spread from the cell membrane into the cell interior. To simplify the analysis, we will further assume that the phosphatase activities

We will first examine numerical solutions of Eq. (8) with the initial conditions

For γ>1 we find a different scenario (see

Examples of the asymptotic, steady state solutions are shown in

The dotted lines in Fig. (A) and (B) are given by Eq. (10). Dashed lines in Figs. (C) and (D) are the exact solutions to Eq. (9) given in the supporting information by Eq. (A2).

In general, there is no simple analytical expression for the stationary solutions of Eq. (8), but we can gain some insight considering the behavior of these solutions within a range where

Importantly, for γ substantially less than 1, we can determine the propagation length for successive activation profiles by using Eq. (9) near the tail of the distribution. Since the approximation _{n}

We can also determine the maximum active concentration (termed the maximal signaling amplitude)

Assuming the same profile shape for the different levels, we can use Eqs. (11) and (12) to estimate the total amount of active component _{n}

In this section we consider a more general case given by Eqs. (5) that allows for saturation kinetics. Following the rescaling as in the previous section, the dynamics of the concentrations is described as follows,

As in the previous section, we consider a cascade with similar properties for all levels and assume

Simulations show that for saturable kinetics, the final steady states are not affected by the initial conditions, similarly as above (^{i}^{a}

To illustrate the threshold behavior of the signal propagation in the parameter space, we consider two cases of large

Dashed lines in Fig A and B are given by the Eqs. (16) and (18), respectively.

For

To obtain an analytical approximation for the boundary between the two regimes of decaying or propagating signals, as in the previous section, we assume that a propagating signal produces a set of stationary concentration profiles with a flat plateau region on the left side of the domain and the concentration

We will first assume that the phosphatases are far from saturation,

When the kinases in the cascade are far from saturation,

Another regime of qualitatively different behavior is the case when both reactions are saturated, i.e. ^{i}^{a}

Cascades of protein modification cycles form the backbone of many signaling pathways, such as MAPK and GTPase cascades, which integrate signals from numerous plasma-membrane receptors and transmit information to distant cellular targets, including the nucleus

The results of the present paper have identified the general conditions for the robust signal propagation and determined when the activation signal stalls near the membrane for an arbitrary number of consecutive layers in a cascade. The signals that spread through a cascade generate a set of stationary activation profiles. When the ratio of the deactivation and activation rates,

Importantly, we found that the control of the spatial signal propagation is dramatically different from the control of transient temporal responses for spatially homogenous cascades

More complex spatial patterns of active kinases are generated when the ratios of phosphatase and kinase activities are different along the cascade (see

In this work we have not considered the effects of feedback and feedforward loops, which may lead to more complex spatial structures and temporal dynamics

It is instructive to compare spatially distributed reaction cascades to other reaction-diffusion systems exploited in mathematical models of biological phenomena. Traveling front or pulse solutions in reactions with multiple steady states or with excitable dynamics (e.g., the Hodgkin-Huxley model) produce concentration distributions that propagate in space with a constant speed, but rarely generate heterogeneous spatial structures at steady states

Exact solutions for small concentrations.

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Temporal evolution of the active form concentration profiles for the sixth level, c6(x), obtained by numerical integration of Eq. (8) with γ = 0.1 and initial conditions for different values of ν at times: (A) t = 5; (B) t = 10; (C) t = 50; (D) t = 10^{4}.

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Temporal evolution of the active form concentration profiles c6(x) obtained by numerical integration of Eq. (8) with γ = 0.1 and ν = 1 for different initial conditions at times: (A) t = 1; (B) t = 10; (C) t = 170; (D) t = 10^{4}.

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Active form concentration profiles, cn(x), obtained numerically from Eq. (8) with γ = 10 at different times: (A) t = 0.01; (B) t = 0.1; (C) t = 1; (D) 10^{4}.

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(A) Stationary total concentration as a function of n for small values of γ. The dashed lines are the linear fit to the data. (B) The slope p as a function of γ obtained by linear fit to the data. The dashed line represents p = 1.3(1−γ)ln(1/γ). The inset in B shows the same figure but with the x-axis in the logarithmic scale.

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Temporal evolution of the active form concentration profiles c6(x) obtained by numerical integration of Eq. (13) with γ = 0.1, ν = 1, ma = 1 and mi = 1 for different initial conditions at times: (A) t = 1; (B) t = 10; (C) t = 170; (D) t = 10^{4}.

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Stationary concentration profiles, cn(x), obtained by numerical integration of Eq. (5) far from saturation with cn(x,t = 0) = 0, D = 1, kn^{a} = 1 for all n, k4^{i} = 10, and kn^{i} = 0.1 for n = 1,2,3,5,6.

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