Biophysics Program, Stanford University, Stanford, CA, USA

Department of Bioengineering, Stanford University, Stanford, CA, USA

Abstract

Background

Bacteria dynamically regulate their intricate intracellular organization involving proteins that facilitate cell division, motility, and numerous other processes. Consistent with this sophisticated organization, bacteria are able to create asymmetries and spatial gradients of proteins by localizing signaling pathway components. We use mathematical modeling to investigate the biochemical and physical constraints on the generation of intracellular gradients by the asymmetric localization of a source and a sink.

Results

We present a systematic computational analysis of the effects of other regulatory mechanisms, such as synthesis, degradation, saturation, and cell growth. We also demonstrate that gradients can be established in a variety of bacterial morphologies such as rods, crescents, spheres, branched and constricted cells.

Conclusions

Taken together, these results suggest that gradients are a robust and potentially common mechanism for providing intracellular spatial cues.

Background

Morphogen gradients form the basis of development in eukaryotes and particularly in embryos, where the spatial distribution of molecules such as maternal mRNAs can give rise to organism-wide properties, transferring information across several orders of magnitude in space

One paradigm for the establishment of intracellular asymmetries in bacteria that has emerged is asymmetric localization of components in a signaling pathway. The bacterium

The establishment of intracellular phosphorylation gradients has been previously explored computationally and theoretically in the context of eukaryotic cells, with previous results illustrating that a membrane-bound kinase and a cytoplasmic phosphatase can give rise to a steady-state cytoplasmic gradient in a spherical cell, similar to the origins of many morphogen gradients

A great deal of attention has also been devoted to how bacteria sense extracellular gradients through processes such as chemotaxis

Here, we use mathematical modeling of reaction-diffusion systems to probe the biochemical requirements for gradient formation. We also investigate how gradients are affected by physical constraints imposed by other cellular processes such as growth and division. Our results encompass a general description of the spatiotemporal dynamics of a substrate subject to regulation by localized sources and sinks, proteolysis, fluctuating concentrations and growth. We find that localized sources and sinks can produce gradients that are robust to most perturbations. We also consider a number of different cell morphologies and localization phenotypes to mimic biologically relevant morphological changes. Our analysis demonstrates that most typical cell shapes and sizes can support gradient formation, suggesting that this may be a common mechanism for providing intracellular spatial cues.

Results

Physical constraints on the establishment of spatial gradients in bacteria

To elucidate the general kinetic and physical constraints on gradient formation, we first consider a simple model of a rod-shaped cell with a source and a sink localized at opposite ends. We use phosphorylation in bacterial two-component systems as a case study; the same analysis can be applied to a source and sink of any other chemical modification with little or no change. In a two-component system, there is a source or sink for the phosphorylated form of the response regulator, R ∼ P
_{
k
}, and R ∼ P is dephosphorylated at the right boundary (_{
p
}. We first ignore synthesis and degradation, such that the total number of molecules is fixed at R_{tot}. The reaction-diffusion equations describing the dynamics of [R] and [R∼P] are then

where _{
p
}.

Between the two poles, where _{
P
x
} + _{
P
}. Because there are reactions occurring at the two poles, acting as sources and sinks for R and R ∼ P, the slopes _{
P
}need not be zero. The no-flux (mass-conserving) boundary conditions at the two poles impose the constraints

which are equivalent to _{
k
}
_{
P
} = _{
p
}(_{
P
}
_{
P
}) = − _{
P
}, indicating that [R] and [R∼P] have opposite slopes and that their sum is a constant _{
P
}independent of _{
P
} by the ratio _{
P
}/

Using the constraint on the total number of molecules

the steady-state solutions can be written in terms of the diffusion time scale _{
D
}=^{2} / _{
k
} = 1/_{
k
}(_{
p
} = 1/_{
p
}) as

These equations immediately indicate the strength of the gradient produced for a given _{
k
} and _{
p
}: for a slow source and sink such that _{
k
} + _{
p
}≫_{
D
}, the normalized slope _{
τ
D
}
_{tot} is much less than 1, indicating a weak gradient. In contrast, as _{
k
} + _{
p
} crosses below the diffusive time scale _{
D
}, the slope approaches a maximum value of _{tot}/^{2} with [R](0) =[R ∼ P](_{tot}/_{
k
} and _{
p
} must be less than _{
D
}to produce a substantial gradient. For a 70 kDa protein diffusing in water, the Einstein relation (_{
B
}
^{2} /s, similar to the measured diffusion constant of a maltose-binding protein in ^{2} /s and note that changing _{
D
} to which all kinetic rates should be compared. For ^{2} /s, the time scale for diffusion between the poles of a 2 _{
D
} = 1 s, and so we require the source and sink rates to be faster than 1/_{
D
}∼1/s in order that phosphorylation can outcompete the uniformity produced by diffusion.

To quantitatively compare the density produced by an asymmetrically localized source and sink to other potential gradient-generation mechanisms, we focus on the phosphorylated (assumed to be active) form and define a metric

which has a maximum value of 15 for _{
σ
p
}→_{
k
}, which determines the total levels of R∼P but scales both poles equally and hence does not affect the ratio.

In the analyses that follow, we wish to account for the dimensions of the cell poles and hence we modify the polar functions to be _{
p
} and 0 elsewhere, and _{
p
}and 0 elsewhere. In Figure
_{
k
} and _{
p
} in a cell of length _{
p
} = 0.25 _{
k
}
_{
p
}≫ 1/_{
D
}. The densities flatten out at the poles due to the source and sink activities not being point sources, nevertheless the polar [R ∼ P] ratio _{
k
}(Figure

A phosphorylation gradient can be produced by fast asymmetric source and sink activities

**A phosphorylation gradient can be produced by fast asymmetric source and sink activities.****A**) Schematic of localization of source and sink. **B**) Mathematical modeling of the spatial asymmetry in phosphorylated response regulator for different source and sink rates. A substantial gradient is obtained only when the phosphorylation rate _{σk}and dephosphorylation rate _{σp}are faster than the inverse of the time scale required for diffusion across the cell, 1/_{D} = 2^{2}. We model the cell in 1D with length _{p} = 0.25 ^{2}/_{k}

Effects of localized synthesis and degradation on gradient formation

Given that changes in activity of the source or sink can enhance or reduce the gradient, we next explored the extent to which synthesis and degradation can affect a gradient. Since dephosphorylation and degradation have similar effects on the R ∼ P density, we sought to address whether localized proteolysis could enhance the gradient by selectively removing protein. We assume that proteolysis of R and R ∼ P occurs at a rate

If synthesis is uniform (

The steady-state solutions for
_{tot}; we will address this saturation below.

Although polarly localized proteolysis could selectively deplete [R ∼ P] at the pole with the higher concentration, we have previously shown that the same constraints determined for phosphorylation kinetics in Sec. 1.1 also apply to localized proteolysis
_{
D
}; in that case, the total response regulator protein levels would show a gradient in addition to the gradient of R ∼ P (Figure

Localized degradation does not significantly affect to the R ∼ P gradient even at high proteolysis rates

**Localized degradation does not significantly affect to the R ∼ P gradient even at high proteolysis rates.****A**) Schematic of the localization of source and sink. Synthesis occurs throughout the cell, whereas degradation occurs only at the phosphatase pole. **B**) Mathematical modeling of the effects of proteolysis on the distributions of R ∼ P for _{D}. (Left inset) The total amount of substrate is not uniform. (Right inset) The ratio of R ∼ P between the two poles is independent of the synthesis rate.

Effects of enzyme saturation on gradient formation

If the levels of the substrate are considerably higher than the levels of the source and sink, the saturation of their activities could effectively decrease the rates of phosphorylation and dephosphorylation and thereby affect gradient establishment. To address the amount of saturation required to perturb a gradient, we modified the rates of phosphorylation and dephosphorylation to reflect Michaelis-Menten kinetics through the effective rates _{
k,e
},_{
p,e
}:

where _{
k
}(_{
p
}) is the unsaturated enzymatic rate, [K_{0}][P_{0}] is the polar kinase (phosphatase) concentration and _{
m
}is the substrate concentration at which the reaction rate is half of _{
k
}[K_{0}](_{
p
}[P_{0}]). We choose _{
m
}/[R_{tot}] = 2 to consider a regime of high sensitivity to the substrate concentration at the pole.

Unlike _{[R ∼ P]phos}); the following arguments are similarly applicable to considerations of [R] at the kinase pole. [R ∼ P]_{phos} levels vary strongly with _{
p
} and less so with _{
k
} (Figure

Source and sink saturation does not significantly reduce the R ∼ P gradient except in very stringent saturation conditions

**Source and sink saturation does not significantly reduce the R ∼ P gradient except in very stringent saturation conditions.****A**) Schematic of localization of source and sink. **B**) Mathematical modeling of the R ∼ P concentration near the phosphatase pole [R ∼ P]_{phos}normalized to the total response regulator concentration [R_{tot}] for varying _{p}, and _{k} = 10 and 100/s. **C**) The effective phosphatase rate _{p,e}is high enough to outcompete diffusion except when the number of enzymes is less than 10% than of the substrate.

Figure
_{
p
} = _{P0} (normalized by the total number of response regulator proteins R_{tot}) and _{
p
}. At high P_{0} levels compared with R_{tot}, the levels of R ∼ P_{phos
} are too low to saturate the enzyme, and _{
p,e
}≃_{
p
}[P_{0}]. As long as the amount of enzyme is in excess of 10% of that of the substrate (P_{0}/R_{tot} > 0.1), a gradient can be established, since the effective phosphatase rate exceeds the diffusion rate.

Effect on delocalized activities on gradient formation

Thus far we have assumed that the substrate is phosphorylated and dephosphorylated only at the poles by its specific partner source and sink, respectively. However,

In the absence of polar activity (_{
k
}=_{
p
}=0), the nonspecific activities define the steady-state levels [R]_{0}and [R ∼ P]_{0} such that

The addition of crosstalk to the polarly localized activities in Figure
_{
k
} = _{
p
}=100/s modified the shape of the steady-state concentration profiles, with a more concave or convex shape if the source or sink crosstalk was dominant, respectively (Figure
_{
k
} and _{
p
}, the localized sink and source ensured that a gradient is maintained. For extremely rapid uniform source activity, the [R ∼ P] profile was somewhat flattened, but the polar ratio

Background source and sink activities do not significantly perturb the R ∼ P gradient even for high rates

**Background source and sink activities do not significantly perturb the R ∼ P gradient even for high rates.****A**) Schematic of localization of source and sink activities; the green and red shading represent the background activity. **B**) Mathematical modeling of the effects of background source and sink activities on the distributions of [R ∼ P]. (Inset) The ratio of [R ∼ P] between the two poles is not significantly affected by background source and sink rates.

Effect of cell length on gradient formation

As a cell elongates, the boundary conditions on the reaction-diffusion equations in Eqs. 1,2 change. Given that the establishment of a gradient is dictated primarily by the relative comparison of the diffusive time scale between the two poles (_{
D
}∼^{2}/2D) and the source/sink rates, we expect gradient establishment to be facilitated by the quadratic increase in _{
D
} as _{
k
}or _{
p
} to 1/_{
D
}will increase by a factor of 4, expanding the regime of rates that satisfy _{
k
},_{
p
} ≫ 1/_{
D
}.

For sufficiently fast _{
k
},_{
p
}(100/s), the polar ratio _{
k
},_{
p
}(1/s), cell growth strongly increases the absolute difference in R ∼ P concentration between the poles. As seen in Figure

Elongation enhances spatial asymmetry

**Elongation enhances spatial asymmetry****A**) Schematic of localization of source and sink. **B**) Mathematical modeling of the effects of varying cell length on the distribution of [R ∼ P] for _{k} = _{p} = 100/_{d}. (Inset) The ratio of [R ∼ P] between the two poles increases linearly with increasing cell length. **C**) At enzymatic rates close to the diffusion rate (_{k}=_{p}=1/_{d}), the difference in [R ∼ P] between the poles increases with cell length; the polar ratio also increases linearly with cell length however it is not affected as significantly as the difference between the poles (inset).

Thus, in physiological conditions, once a gradient is established, cell elongation enhances asymmetry in R ∼ P. Moreover, for small _{
k
}or _{
p
}, there might not be a significant gradient until the cell grows past a length _{
D
} can be overcome by the source and the sink.

Effect of cell geometry on gradient formation

Thus far we have focused on one-dimensional diffusion to illustrate the factors that influence gradients in rod-shaped cells. However, the three-dimensional geometry of cells could potentially impact the consequences of an asymmetric source and sink localization pattern. In addition to rod-shaped species such as

To keep our analysis consistent with the 1D modeling discussed above we maintained the diffusion constant ^{2}/s and distributed the kinase and phosphatase activities on the cell membrane at opposite poles, maintaing a constant active surface area across all 3D simulations. We also appropriately scaled the relevant enzymatic activities _{
k
} and _{
p
}, such that the total number of active kinases and phosphatases remained consistent. This adaptation of the kinase and phosphatase rates to 3D allows for a direct comparison of all geometries and across dimensions.

Gradient formation in a crescent-shaped cell

Several model organisms such as _{
c
}and assumed that the sink and source were oppositely localized at the hemispherical poles (Figure
_{
c
} = 1

Cell bending does not significantly affect the R ∼ P gradient even for small cellular radius of curvature

**Cell bending does not significantly affect the R ∼ P gradient even for small cellular radius of curvature.****A**) Schematic of a bent cell with radius of curvature _{c}with oppositely localized kinase and phosphatase. **B**) 3D mathematical modeling of the distribution of [R ∼ P] in a bent cell with radius of curvature _{c} = 1 **C**) [R ∼ P] line scan through the cell middle showing volume-weighted average density along cell length (along the dashed line in A) for different _{c.}

Gradient formation in round cells

Many important model bacteria, such as

Spherical cells can support gradients along the source/sink axis

**Spherical cells can support gradients along the source/sink axis.****A**) Schematic of a spherical cell of radius _{s}with oppositely localized kinase and phosphatase spread over a spherical cap. **B**) 3D mathematical modeling of the distribution of [R ∼ P] in a spherical cell with radius _{s} = 0.4 **C**) [R ∼ P] line scan through the cell middle showing average volume-weighted density along the cell length (along the dashed line in **A**) for different radii _{s}. **D**) Schematic of 1D distributions of kinase and phosphatase activities that produce profiles mimicking those in (**C**) for _{s}> 0.5

Using the understanding generated by our 1D simulations, we noted that the flattening of the density near the equator was similar to the effect of background kinase and phosphatase activities in Figure

Gradient formation in disk-shaped cells

Rod-shaped bacteria such as _{1} between 0.2 and 1.25 _{2} = 1_{1} >=_{2} and to the rod-shaped cells (Figure
_{1} < _{2} (Figure

Disk-shaped cells can support gradients along the source/sink axis

**Disk-shaped cells can support gradients along the source/sink axis.****A**) Schematic of a disk-shaped cell with distance _{2} = 1 _{1} between 0.2 and 1.25 **B**) 2D Mathematical modeling of the distribution of [R ∼ P] in an ellipse geometry. **C**) [R ∼ P] line scan through the cell middle showing volume-weighted average density along cell length (along the dashed line in A) for varying ellipticity.

Gradient formation in branched cells

Several bacterial species grow as branched rods in the absence of particular nutrients or in response of environmental cues. In particular the bifid, or Y-shaped morphology, is common in species such as _{
b
}). Similar to simulations involving a spherical cell with large radius (Figure

Spatial asymmetry in [R ∼ P] is not sensitive to branching

**Spatial asymmetry in [R ∼ P] is not sensitive to branching.****A**) Schematic of a branched cell with oppositely localized kinase and phosphatase in the rod-shaped region of the cell. The branch occurs at midcell at a right angle to the rest of the cell, and has length _{b}. **B**) 3D mathematical modeling of the distribution of [R ∼ P] in a branched cell with _{b} = 0.5 **C**) [R ∼ P] line scan through the cell middle showing volume-weighted average density along cell length (along the dashed line in A) for varying branch lengths. **D**) Schematic of 1D distribution of kinase and phosphatase activities that produce profiles mimicking those in (**C**) for _{b} > 0.5

Similar to simulations in spherical cells (Figure

Therefore, even the growth of a branch comparable in length to the rest of the cell length does not significantly affect gradient formation.

Gradient maintenance during cell division

During cell division, the septum provides a barrier to diffusion. For Gram-negative rod-shaped organisms such as

To investigate how the constricted morphologies during division affect gradients, we performed simulations in which we varied the fraction _{
a
} of the cross-sectional area of the cell that is encompassed by the pore at the constriction site between two dividing cells (Figure
_{
a
}decreases from 1 to 0 during division. We found that the gradient was not noticeably altered until the constriction was significant (_{
a
}≲0.3) (Figure
_{
a
} was very close to zero.

Spatial gradients are sensitive to cell constriction only when the septum provides a significant diffusion barrier

**Spatial gradients are sensitive to cell constriction only when the septum provides a significant diffusion barrier.****A**) Schematic of a dividing cell with oppositely localized kinase and phosphatase. _{a}is the ratio of the area of the pore at the constriction site to the cross-sectional area in the cylindrical portion of the cell. **B**) 3D mathematical modeling of the distribution of [R ∼ P] in a dividing cell with _{A}=0.3. **B**) [R ∼ P] line scan through the cell middle showing volume-weighted average density along cell length for varying constriction sizes. **D**) Schematic of 1D distribution of kinase and phosphatase activities that produce profiles mimicking those in (**C**) for _{A} < 0.1

We noted that the R ∼ P distribution in Figure
_{
A
} < 0.3 can be recapitulated in 1D simulations by adding a source and a sink adjacent to the left and right of the constriction site, respectively (Figure

Discussion

Spatial asymmetry in bacterial cells is often cell-cycle regulated and highly dynamic, and a natural consequence is the production of spatial gradients. We have established that gradients produced by a localized source and sink will be robust and significant as long as the kinetics of the source and sink are on timescales faster than the typical time required to diffuse across the length of the cell (Figure

Just as gradient formation requires fast enzyme kinetics, in order for other biochemical processes to disrupt an existing gradient, their kinetics must be similarly fast. In our model, fluctuations in enzyme concentrations can be directly mapped to changes in the effective enzymatic rates. Therefore, as shown in Figure

Our modeling also applies to pathways besides two-component systems, such as the synthesis of the cytoplasmic second messenger cyclic-di-GMP by diguanylate cyclases and degradation by phosphodiesterases. While fast kinetics are required for gradient formation, localized kinases and phosphatases can also give rise to asymmetries after cytokinesis, solely by segregating localized components. For example, cyclic-di-GMP is asymmetrically distributed in

While cyclic-di-GMP asymmetry appears to require cell division, other bacterial pathways rely on fast kinetics of localized kinases and phosphatases to establish spatial gradients prior to division. For example, DNA replication in

We have also demonstrated that a representative sample of bacterial cell shapes and sizes can support gradient formation through oppositely localized kinase and phosphatase activities (Figures

Conclusions

Our analysis of gradient formation encompasses a wide range regulatory mechanisms and morphologies, demonstrating the conditions under which robust spatial gradients can be realized for providing intercellular spatial cues. Our results highlight the utility of mathematical modeling in future studies of intracellular organization in bacteria, and illustrate the complex spatial patterning that can be achieved even in the absence of membrane compartmentalization.

Methods

Unless otherwise stated, the kinase and phosphatase _{
k
} = _{
p
} = 100/s and the diffusion constant was ^{2}/s. For all 1D and rod-shaped cell simulations, the cell length was

The steady-state solutions in 3D geometries were determined using an in-house, custom-written Matlab (The Mathworks, Inc., Natick, MA, USA) software package called TURING, developed for simulating reaction-diffusion equations in complex geometries. TURING includes a graphical interface for creating finite-element grids representing biologically relevant morphologies, and a symbolic library capable of interpreting intuitively defined reaction-diffusion equations and parameters to build a model for a simulation. TURING solves the system of reaction-diffusion equations on the specified grid with a fully implicit, numerical method. The source and sink elements reside on the cell membrane, occupying the same surface area _{
s
} as a pole in the case of a rod-shaped cell with radius _{
p
} = 0.25 _{
s
} = 0.39 ^{2} for both kinase and phosphatase elements. The enzymatic rates _{
k
}and _{
p
} were set to 100/s for the curved cylinder geometry. These rates were appropriately scaled for the other geometries, such that the number of active kinases and phosphatases remained consistent between simulations. This corresponded to holding the products _{
k
}
_{
k
} and _{
p
}
_{
p
} constant for all geometries, where _{
k
}and _{
p
} are the total volumes of the kinase and phosphatase finite elements respectively.

Cells with azimuthal symmetry were solved ignoring azimuthal diffusion. Curved cylinder, spherical, and constricting cell geometries were represented by a grid with elements whose average side lengths were 75, 10, and 25 nm, respectively.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

C.T. and K.C.H. designed the study. C.T., N.R. and K.C.H. designed and performed the mathematical modeling. C.T. and K.C.H. wrote the manuscript. All authors discussed the results and commented on the manuscript. All authors read and approved the final manuscript.

Acknowledgements

This work was funded in part by NIH grants K25 GM075000 and an NIH Director’s New Innovator Award DP2OD006466 to K.C.H. C.T. received support from the Stanford Graduate Fellowship and the Bruce and Elizabeth Dunlevie Bio-X Stanford Interdisciplinary Graduate Fellowship. N.R. received support from a Stanford REU summer fellowship.