The dynamics of the concentration c of a species of diffusing molecules is described by the diffusion equation (1)∂c∂t(r,t)=D∇2c(r,t), where r = (x, y, z) denotes the position in space, t parameterizes time, and ▽² is the Laplace operator, (2)∇2=∂2∂x2+∂2∂y2+∂2∂z2.

Consider a point-like emitter which emits an instantaneous pulse of n₀ molecules at time t = 0. The concentration that is measured at a given distance r=x2+y2+z2 from the point of emission after a time t has elapsed, is then (3)G(r,t)=n0(4πDt)32e-r24Dt, which is the fundamental solution (http://en.wikipedia.org/wiki/Fundamental_solution, 2012-11-25) of the diffusion equation (Eq. 1).

We next consider a single water channel in an infinite, flat impermeable membrane. We refer to this geometry as the open geometry. An aquaporin is less than 10 nm wide and is therefore effectively point-size compared to the distances over which the signal is transmitted (1–10 μm). We assume that the channel lets an instantaneous pulse of n₀ molecules cross the membrane. The geometry is the same as above except for the membrane, which restricts the molecules to one half of space. The concentration in that half is hence twice that given in Eq. 3, (4)c0(r,t)=2n0(4πDt)32e-r24Dt.

Equation 4 is the fundamental solution of the diffusion equation in the open geometry with its closed boundary condition at the membrane at x = 0, (5)∂c0(0,y,z),t∂x=0.

Now assume that channels are distributed evenly over the cell membrane. We denote by ρ_a the surface density of these channels. If in an area of size a × a all of these emit simultaneously, then the emitted signal can be found by adding up the concentrations from all emitters in that area, using c₀ for each emitter. That is, we find the concentration as the convolution of the fundamental solution (Eq. 4) with the constant density profile over the emission area.

At a point on the x-axis a distance x from the membrane – i.e., on the normal through the center [(y, z) = (0, 0)] of the area containing channels (Figure 1A) – the measured concentration after emission is thus (7)c(x,t)=N0πDte-x24Dterfa4Dt2, where N₀ = ρ_an₀ and erf denotes the error function, (8)erf(x)=2π∫0xe-s2ds.

Note that when we are far enough away from the source (approximately when x is larger than a), the signal looks the same no matter what the source looks like; the forms of signals from different emission areas are very similar, the only difference is the scale, which is determined by the total number of molecules emitted. The same similarity of signals occurs even close to the emission area at late times, when the initial shapes of all signals have disappeared by diffusion and all signals are described by Eq. 4 (albeit with different amplitudes). This also means that the specific shape of the emission area is not important. Signals emitted from a square, circular, or differently shaped areas all look alike as long as the dimensions of the areas are roughly the same.

For simplicity, we have assumed above that the dimensions of the cell are much larger than a, the linear extent of the area emitting the signal, i.e., the cell is effectively infinitely large (Figure 1A). We refer to this case as the open geometry, because most of the cell membrane is absent, effectively, being too far away to matter in the diffusive dynamics of concentrations of signaling molecules. In the Results section, Figures 3–7 present results for this geometry.

We shall see below that for the signal to be detectable at the target (∼10 μm from the source), the area of emission must be of the same order (∼10 μm × 10 μm) as the dimension of an epidermal cell (∼10 μm × 20 μm × 100 μm). In an extreme example, the whole end-wall (10 μm × 20 μm) of the cell emits a synchronized signal and the cell walls act as closed (reflecting) boundaries (Figure 1B). Because of the closed boundary conditions the concentration inside the cell does not depend on the y- and z- coordinates and this geometry is mathematically equivalent to a point source emitting in one dimension. Then the concentration measured at time t after emission only depends on the distance in the x-direction from the source and on the length L of the cell, e.g., L = 100 μm. It is calculated using the method of images (Griffiths, 1999) and is (9)c(x,t)=∑m=-∞∞N0πDte-(x-2mL)24Dt.

In the Section “Results,” Figures 8–12 present results from such a closed system. Note that while the sum in Eq. 9 in principle is infinite, only a few terms with values of m near 0 contribute in practice.

The refractory time t_ref is the time it takes, after detection of a pulse signal, for the concentration in a given point to drop to a sufficiently low value to allow detection of a new pulse signal. We define this threshold concentration to be some fraction α of the peak concentration c_max attained at that point in consequence of an emitted signal, the trough/peak ratio. This peak concentration occurs at t_max by definition of this time. In the simplest possible case of the point source, this time is equal to the diffusion time, defined as t_D = x²/(6D) in three dimensions. In the simple case of an open geometry, one arrives at the same result, approximately, in cases where the target is further from the source than the size of the emission area. This approximation improves with distance from the source. In a closed geometry, t_max is approximately equal to x²/(2D), which is the diffusion time in one dimension. In general t_max must be found numerically.

The refractory time is found as the solution to the equation (10)c(x,tref)=αc(x,tmax), and is (11)tref=-tDW-α23e in the simple cases of a point source as well as for an open geometry resembling a point source. Here W is the Lambert-W function, defined implicitly by W(x)e^W(x) = x.

An instantaneous pulse is a mathematical abstraction, which is easy to work with, but which does not exist in reality, since an instantaneous emission of a signal of some amplitude N₀ would imply an infinite flux over the membrane. Instead, the channels stay open for a duration Δt during which a flux j₀ passes through, giving a total signal amplitude of N₀ = j₀Δt. In the geometry described by Eq. 7, we cannot solve the diffusion equation analytically, and we must solve the diffusion equation (Eq. 1) numerically. This is done using Comsol Multiphysics, a finite element solver for physics and engineering applications.

However, for short times after emission, in the closed geometry or close to the source in the open geometry (i.e., in a point located at a distance from the source which is much smaller than the spatial extent of the source), the system is mathematically equivalent to that of a point source emitting in one dimension. Furthermore, at long times or far from the source in the open geometry, the system is equivalent to that of a point source emitter in three dimensions. Both these cases can be solved analytically. In the 1D case, for times t ≤ Δt, the solution is (12)ct≤Δt(x,t)=j0D4Dtπe-x24Dt-xerfcx4Dt where erfc is the complementary error function, erfc = 1 − erf. For t > Δt, the solution is (13)ct>Δt(x,t)=ct≤Δt(x,t)-ct≤Δt(x,t-Δt).

In the 3D case, the concentration measured at a distance x from the source along a line perpendicular to the membrane is for t ≤ Δt, (14)ct≤Δt(x,t)=j02πDxerfcx4Dt, and for t > Δt, (15)ct>Δt(x,t)=ct≤Δt(x,t)-ct≤Δt(x,t-Δt).

Figure 3 illustrates these analytical results. As long as Δt is approximately equal to or shorter than the characteristic diffusion time of an instantaneous pulse, the signal does not differ significantly from that of an instantaneous pulse. We refer to a signal of this type as a short pulse signal.

As far as we know, H₂O₂ is mainly transported over the membrane by aquaporins. These are passive channels, so the concentration inside the membrane cannot exceed the concentration c_out outside the membrane. Furthermore, if the concentration inside the membrane approaches the outside concentration, the flux will diminish.

The maximal concentration on the inside of the membrane is found at time Δt and is (from Eq. 12) (16)c(0,Δt)=j04ΔtDπ.

This means that j₀ must be much smaller than πD∕4Δtcout, or equivalently Δt≪πDcout24j02 for the flux to stay constant during the time that the channels are open.

Hydrogen peroxide is degraded by enzymes inside the cell. We assume Michaelis–Menten kinetics. Then the evolution of the concentration is described by the reaction-diffusion equation (17)∂c∂t(r,t)=-Vmaxc(r,t)Km+c(r,t)+D∇2c(r,t).

The reaction-diffusion equation is non-linear, and no known analytical solution exists. We solve the equation numerically using Comsol.

We also consider a harmonically oscillating signal in the closed geometry, that is, a harmonically oscillating flux over the end membrane (x = 0) of the cell. This is described mathematically by the boundary condition (18)∂c∂x(0,t)=j0D(1-cos2πft), where j₀ is the average flux and f is the frequency of the signal. We solve the diffusion equation with degradation (Eq. 17) in one dimension numerically using Comsol with the boundary condition given by Eq. 18 at x = 0 and a closed boundary at x = 100 μm.

We investigate the effect that cytoplasmic streaming may have on signaling. A cytoplasmic stream can be modeled as a roughly cylindrical stream of infinite extent in the open geometry (Figure 2A). If a signaling molecule diffuses into this stream, it will be translated in the direction of the stream with a speed approximately equal to the average speed v found in a cross-section of the stream until the molecule diffuses out of the stream again. As Figure 2A shows, molecules can circumvent the stream and many will reach the target without having crossed the stream. Those whose do cross it will spend an amount of time in the stream which is on the order of the characteristic diffusion time, (19)tD=d24D where d is the width of the stream or the distance between the source and the target, whichever is smaller. During this time, a particle in the stream is translated a distance (20)Δy=vd24D=Pe4d in the direction of the stream (the y-direction). Here Pe is the Péclet number. In biological cells Pe is typically much smaller than one.

Since d is smaller than the distance from the source to the target, Δy is negligible compared to the source-target distance if Pe is smaller than one. Thus, as long as Pe is smaller than one, the effect of cytoplasmic streams is negligible.

In the previous subsection, we showed that cytoplasmic streaming has a negligible effect on signal transmission via diffusion in plant cells, by using an order-of-magnitude argument. The present subsection provides a more rigorous proof. It is rather equation-heavy and can be skipped.

We consider a cytoplasmic stream which fills the entire inside of the “cell” in the open geometry, moving with velocity v everywhere (Figure 2B). This is an extreme case; it is not physically realistic, but any realistic stream will do less transport of signaling molecules. So if we can show that this stream only has negligible effect on the signal measured at the target, then we know that a more realistic cytoplasmic stream, smaller and roughly cylindrical, e.g., has an even smaller effect.

At time t after emission of a short pulse signal from a point-like emitter located at (x,y,z) = (0,0,0), the concentration at the point r inside this cell is equal to (21)c(r,t)=2n0(4πDt)32e-x2+(y-vt)2+z24Dt.

The time t_max when the concentration reaches its peak at the point r, is found by setting the derivative of Eq. 21 to zero and solving for t. This gives (22)tmax=rvζ, where (23)ζ=3Pe2+1-3Pe and Pe is defined in Eq. 20. Measured at a distance x from the source and perpendicular to the stream (y, z) = (0, 0), the peak signal is then (24)cmax(x)=e-Pe1+ζ24ζ4πDxζv32.

This should be compared to the peak signal in the absence of a cytoplasmic stream, (25)c0,max(x)=e-322πr2332.

Thus, the cytoplasmic stream decreases the peak concentration by a factor of (26)cmax(x)c0,max(x)=Pe6ζ32e32-Pe1+ζ24ζ.

Even for a Péclet number of one, corresponding to a stream speed of 100 μm s⁻¹ (the highest reported in literature) and a source-target distance of 10 μm (the longest considered), this factor is equal to 0.96. The cytoplasmic stream only decreases the peak signal by 4% in this extreme case. Furthermore, in this case the time at which the peak concentration is measured is 0.97 times t_D, the time at which the peak concentration is measured in the absence of cytoplasmic streaming. So we can conclude that the effect of cytoplasmic streaming on the signal is completely negligible in more realistic, less extreme, scenarios.

The values of the various parameters introduced above and the way they influence the hydrogen peroxide signal, will be treated stepwise in the Results section. Table 2 shows typical values of these parameters.

H₂O₂ crosses the PM through aquaporins. The maximal flux through a single aquaporin channel is on the order of 10⁹ molecules channel⁻¹ s⁻¹ (Jensen and Mouritsen, 2006) while the number of channels per μm² is on the order of 30 (Li et al., 2011). This gives a maximal flux per μm² of 3 × 10¹⁰ molecules μm⁻² s⁻¹, or in units of moles: 5 × 10⁻¹⁴ mol μm⁻²s⁻¹. If we assume that the aquaporins allow H₂O₂ and water molecules to pass with rates proportional to their respective concentrations and we further assume that the concentrations outside the PM are 100 mM for H₂O₂ (very high) and 55 M for water (standard), then the maximal flux of H₂O₂ over the cell membrane is 10⁻¹⁶ mol μm⁻² s⁻¹.

A prerequisite for attaining this maximal flux is that the concentration of H₂O₂ outside the membrane is much higher than the concentration inside, since the aquaporins are passive channels. This is, however, always the case in practice, since even if we let the channel stay completely open for 1 s, the concentration will maximally reach 4 mM inside the cell.

A signal is emitted at the PM by opening the aquaporins for a while. We first consider signals consisting of short pulses. All other types of signals can be constructed by adding short pulses and they are the type of signal, which allows the fastest frequency modulation. A short pulse is obtained by fast opening and closing of the channels in the membrane. For mathematical convenience this can be idealized to an instantaneous pulse where the channels are opened and closed infinitely fast and the flux over the membrane is infinite during this infinitesimal time-interval.

Increasing the time the channels stay open naturally increases the signal amplitude. At long times after emission, there is a simple proportionality between the time the channels stay open (and thus the total signal) and the concentration measured at the target; as with the different emission areas the curves describing the concentration all collapse to the same on curve (Figure 3), except for a scaling factor determined by the total amount of H₂O₂ emitted. That is, at sufficiently long times the curves become equal except for a scaling factor.

For times shortly after the emission of the signal, it is another story. Longer opening time flattens out the peak of the concentration, and the peak concentration, relative to the concentration measured at long times, is decreased. This in turn increases the refractory time.

When we consider a target that is 1 μm away, the signal can be considered a short pulse as long as the time during which the aquaporins stay open is approximately 1 ms or shorter, since its relative amplitude does not differ significantly from that of an instantaneous pulse (Figure 3A). The maximal amount of H₂O₂ that can be released in a single short pulse per unit area, i.e., the maximal amplitude of a short pulse signal, is then 10⁻¹⁹ mol μm⁻². At a distance of 10 μm, the signal can be considered a short pulse, if the aquaporins stay open for approximately 10 ms or less (Figure 3B), i.e., the maximal amplitude of such a short pulse is 10⁻¹⁸ mol μm⁻². Here, where the source-target distance (10 μm) is comparable to the dimensions of the source (10 μm × 10 μm), we also see that the signal received at the target is well approximated by the analytical solution for a point source (Figure 3B).

The size of the activated area on the PM, the area creating the synchronized signal, may vary depending on, e.g., what caused the signal. This gives different signals. We consider emission area sizes ranging from a single channel to 100 μm²: (i) a single channel, (ii) 1 μm × 1 μm (a small patch), and (iii) 10 μm × 10 μm (a large patch extending over a major fraction of one side of a plant cell).

Varying the area of the patch from where molecules are emitted, affects the total amplitude of the signal, if other parameters are kept constant (Figure 4). Furthermore, a larger emission area means slower decay, i.e., longer refractory time (Figures 5 and 6).

For a signal emitted by a single channel, it is clear that we would need a very high initial signal amplitude to be able to detect the signal, even as close as 1 μm away. If we assume that the resting concentration of H₂O₂ is 0.1 μM and that we can detect the signal at a concentration 10-fold higher than this, we need a signal twenty times larger than what a single channel can emit in a short pulse, to be able to detect the signal from a single channel at 1 μm from the source. Thus, in the following we will only consider emission from sources of size 1 μm × 1 μm and 10 μm × 10 μm.

The refractory time does not depend on the signal amplitude, but depends highly on distance from the source and emission patch size (Figures 5 and 6). Increasing the distance between the source and the target increases the refractory time and, additionally, drastically decreases the amplitude of the signal (Figures 5 and 6). In the open geometry the main obstacle to signaling is the difficulty of attaining a peak concentration at the target which is sufficiently high to be detected.

At a distance of 1 μm (Figure 5), emission of a pulse of 10⁻¹⁹ mol μm⁻² from an area of 1 μm × 1 μm (total signal is 10⁻¹⁹ mol) creates a detectable signal with a refractory time of 2.3 ms for a peak/trough ratio of 10 (11 ms for a ratio of 100), indicating that in the open geometry frequencies of up to 500 Hz are feasible even without enzymatic degradation of H₂O₂. When the area is larger, 10 μm × 10 μm (total signal: 10⁻¹⁷mol), the signal is considerably stronger, but the refractory time increases to 19 ms (100 ms for a peak/trough ratio of 100), indicating that here frequencies up to 50 Hz are possible.

With distance, the wave broadens, thereby increasing the refractory time. At a distance of 10 μm (Figure 6), only an emission from the largest area considered (10 μm × 10 μm) is detectable, even if the detection limit is as low as 1 μM. Here the refractory time is 0.22 s for a peak/trough ratio of 10 (1.1 s for a ratio of 100), indicating that signaling with frequencies up to 5 Hz is feasible.

In general, as long as the distance between the target and the source is greater than or equal to the dimensions of the area of emission, the signal is well approximated by the signal from a point source (compare Figures 5A,B and 6A–C) and the refractory times are given approximately by the analytical result for a point emitter (Eq. 11). In this case, for a source-target distance of 1 μm, the refractory times for the signal to reach 1/10th and 1/100th of its peak value for a point source are 1.9 and 9.6 ms respectively. For a source-target distance of 10 μm, the refractory times for a point source are 0.19 and 0.96 s.

Figure 7 shows what a train of short pulses emitted from a 10 μm × 10 μm area looks like at a distance of 10 μm from the source. At a low frequency, say 0.1 Hz, the H₂O₂ concentration has time to return to the resting level (Figure 7A), at 1 Hz, the level does not quite return to 1/100th of the peak value since the refractory time is 1.1 ms (Figure 7B), and at a frequency of 10 Hz there is a build-up of H₂O₂ and only a twofold difference between peak and trough, insufficient for the signal to be deciphered (Figure 7C).

We have until now assumed that the cell is so large that there is no effect of cell walls on the distribution of H₂O₂, except from the wall through which it enters the cell. In reality, and especially when we look at repeated pulses and other cases of continuous signaling, H₂O₂ will build-up in the cytosol and thereby prevent further signaling, unless it is removed.

We consider a closed geometry in which the signal is emitted from the whole end-wall of the cell, e.g., an area of 10 μm × 20 μm. This is the most restricted geometry that one can imagine, i.e., it complements the open geometry treated above and thereby delimits the range of possible phenomena. This is the geometry in which one can have the highest signal amplitudes, but on the other hand also the longest refractory times.

If no H₂O₂ is removed or degraded, signaling over 10 μm is not possible since the cell fills up with H₂O₂ and the concentration settles at a constant value throughout the cell of approximately 20% of cpeak as measured at 10 μm from the source. However, there are several mechanisms by which H₂O₂ can be removed – transport out of the cell or into organelles, chemical reactions, and enzymatic reactions. We cannot evaluate the magnitude of the former two and if transport out of the cell is passive it will counter the effect of the cell walls to some extent, i.e., the system will be described by the open geometry or some intermediary between the closed and open geometries. So here we will let H₂O₂ removal consist entirely of enzymatic degradation to analyze how H₂O₂ removal affects signaling. The cytosol contains several enzymes capable of removing hydrogen peroxide, e.g., ascorbate peroxidase. When we include removal of H₂O₂ by Michaelis–Menten kinetics in the model, we see that signaling over 10 μm becomes possible at frequencies up to 2.5 Hz (for a peak/trough ratio of 10) when the removal rate is 100 μM s⁻¹ (Figures 8D and 9A). At a much lower removal rate (1 μM s⁻¹) only signaling at rates up to 0.06 Hz is possible (for a peak/trough ratio of 10, 0.01 Hz for a ratio of 100, Figure 8C). Since the diffusion of H₂O₂ is fast on intracellular length scales, degradation does not affect the maximal concentration of H₂O₂ much (Figure 8), though it significantly decreases the refractory time. At an intermediate removal rate (10 μM s⁻¹), similar to the one reported by Bonifacio et al. (2011) signaling at 0.2 Hz is possible (Figures 9B,C). In all cases peak concentrations of 50 μM at the target can be reached.

The Michaelis constant K_m may change as well, i.e., due to molecular crowding (see Effect of Molecular Crowding). We have considered a ten-fold increase or decrease in K_m (i.e., to 200 or 2 μM) and see that the effect of an increase in K_m is qualitatively the same as the effect of a corresponding decrease of V_max and vice versa (Figures 8E,F).

Frequencies, amplitudes, and background H₂O₂ concentrations, which make signaling possible are listed in Tables 3 and 4.

Thanks to the late French physicist Joseph Fourier (the inventor of Fourier analysis), we know that any signal can be constructed as a sum of harmonic functions, i.e., sines and cosines. This type of signal is complimentary to the short pulse, since it is regulated gradually, whereas the short pulse is regulated by an abrupt open/close mechanism. The manner in which such a signal propagates thus gives us a new angle on the problem of signal transmission. This angle is the natural one to use in an investigation of the propagation of pulse trains, because a harmonic signal is the simplest possible periodic pulse train as seen from a mathematical point of view, and because more complicated pulse trains can be viewed as super-positions of these simple ones for analytical purposes. That is the essence of Fourier analysis. Add to this that harmonic signals are the carrier waves for signals encoded by frequency modulation.

While the non-linear kinetics of the Michaelis–Menten degradation term prevents us from solving the kinetic equation using Fourier analysis, that analysis still applies to the emitted signal. It is the interactions of different Fourier components of the propagating signal in the degradation term that bars Fourier analysis.

Figure 10 shows what a signal with a harmonic source looks like throughout the cell at different points in the emission cycle, i.e., snapshots of the concentration profile at different points in time. No spatial wave-like pattern is seen. This is a signature feature of diffusive transport in a non-excitable medium. This is because signal propagation is only driven by concentration gradients and the concentration of signaling molecules thus always will be highest at the source. The wave-pattern is only seen in the time-domain, e.g., when we follow the concentration in a single point over time (Figure 11). To see spatial wave-patterns in diffusion-mediated signaling a non-linear, excitable medium is required.

Though harmonic signals in theory allow much higher signaling amplitudes, they are limited by the refractory time, and the possible peak concentrations are lower for harmonic signals than for short pulse at similar frequencies (Figure 11; Table 5). Figures 12A,B shows possible signaling frequencies as a function of V_max for signaling using short pulses (Figure 12A) and a harmonically oscillating flux (Figure 12B).

Molecular crowding effectively lowers the diffusion coefficient of a molecular species in the cytosol compared to its value in water. For calcium ions in animal cells, this effect is of the order of 20–50% (Straube and Ridgway, 2009). We have assumed a similar effect on the diffusion coefficient of H₂O₂ in plant cells, lowering the diffusion coefficient of H₂O₂ from 1,700 μm² s⁻¹ (van Stroe-Biezen et al., 1993) to 1,000 μm² s⁻¹. Excluding the effect of degradation, a twofold increase of the diffusion coefficient will simply decrease the characteristic diffusion time and refractory times by a factor two, but will not influence signal amplitudes; an increase of the diffusion coefficient diminishes the influence of degradation, since it reduces the time-scale of signaling.

Molecular crowding may also affect the binding rates and rates of catalysis of enzymes and thus change K_m and V_max (Zhou et al., 2008). A change of the rate of catalysis changes both K_m and V_max in the same direction, which means that these two effects largely cancel each other in this case (Figure 8). A change in the association rate between signaling molecules and the degradation enzymes only affects K_m and will thus to a larger extent affect reaction kinetics. The effect of crowding varies highly for different enzymes (Norris and Malys, 2011) and is unknown for enzymes degrading H₂O₂ in plant cells. So we have considered a wide range of possible values for K_m (2–200 μM, Figure 8).

Finally, we have investigated what effect cytoplasmic streaming may have on signaling. We have considered a roughly cylindrical cytoplasmic stream in between the source and the target (Figure 2A). A simple order-of-magnitude argument showed that the influence of the stream is negligible if the Péclet number is on the order of one or smaller. Here the Péclet number is defined as the ratio between the stream’s width times the stream velocity and the diffusion coefficient. Considering the maximal stream speed reported in the literature, v = 100 μm (Goldstein et al., 2008; Table 2), and considering a stream which is as wide as the largest distance between the source and the target, 10 μm, we get a Péclet number of one. This means that even in this extreme case, cytoplasmic streaming does not significantly disturb signaling.

Thus, cytoplasmic streaming does not hinder intracellular signaling by diffusion, since the diffusion coefficient is so large compared to typical values of the cytoplasmic stream width and speed (Table 2). On the contrary, it lowers the refractory time, because the stream has a stronger delaying effect on the signal molecules that arrive late at the target than it has on the early arrivers, due to the former spending more time in the stream than the latter. The net effect is a sharper profile of concentration plotted against time, similar to the situation in which signal molecules are degraded.