Modelling bacterial chemotaxis for indirectly binding attractants

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Modelling bacterial chemotaxis for indirectly binding attractants Pei Yen Tan , Dr. Marcos Marcos , Yu Liu PII: DOI: Reference:

S0022-5193(19)30489-8 https://doi.org/10.1016/j.jtbi.2019.110120 YJTBI 110120

To appear in:

Journal of Theoretical Biology

Received date: Revised date: Accepted date:

17 July 2019 9 November 2019 16 December 2019

Please cite this article as: Pei Yen Tan , Dr. Marcos Marcos , Yu Liu , Modelling bacterial chemotaxis for indirectly binding attractants, Journal of Theoretical Biology (2019), doi: https://doi.org/10.1016/j.jtbi.2019.110120

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HIGHLIGHTS   

Indirect binding expression for attractants mediated by periplasmic binding protein Simulations for maltose response agree well with experimental data from literature Scaling factor for attractant concentration from extracellular to periplasmic space Predicting AI-2 response may require further refinement to existing model

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Modelling bacterial chemotaxis for indirectly binding attractants Pei Yen TAN1, Marcos2*, Yu LIU1,3 1

Advanced Environmental Biotechnology Centre, Nanyang Environment & Water Research

Institute, Nanyang Technological University, 1 Cleantech Loop, Singapore 637141; 2School of Mechanical and Aerospace Engineering, Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798; 3School of Civil and Environmental Engineering, Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798

*Corresponding author: Tel.: +65-67-905-867 E-mail: [email protected]

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Abstract In bacterial chemotaxis, chemoattractant molecules may bind either directly or indirectly with receptors within the cell periplasmic space. The indirect binding mechanism, which involves an intermediate periplasmic binding protein, has been reported to increase sensitivity to dilute attractant concentrations as well as range of response. Current mathematical models for bacterial chemotaxis at the population scale do not appear to take the periplasmic binding protein (BP) concentration or the indirect binding mechanics into account. We formulate an indirect binding extension to the existing Rivero equation for chemotactic velocity based on fundamental reversible enzyme kinetics. The formulated indirect binding expression accounts for the periplasmic BP concentration and the dissociation constants for binding between attractant and periplasmic BP, as well as between BP and chemoreceptor. We validate the indirect-binding model using capillary assay simulations of the chemotactic responses of E. coli to the indirectly-binding attractants maltose and AI-2. The predicted response agrees well with experimental data from a number of maltose capillary assay studies conducted in previous literature. The model is also able to achieve good agreement with AI-2 capillary assay data of one study out of two tested. The chemotactic response of E. coli towards AI-2 appears to be of higher complexity due to reports of variable periplasmic BP concentration as well as the low concentration of periplasmic BP relative to the total receptor concentration. Our current model is thus suitable for indirect binding chemotactic response systems with constant periplasmic BP concentration that is significantly larger than the total receptor concentration, such as the response of E. coli towards maltose. Further considerations may be taken into account to model the chemotactic response towards AI-2 with greater accuracy.

3

1. Introduction Single-celled organisms such as bacteria navigate their surroundings through the use of simple yet sophisticated sensory systems, responding to various positive and negative stimuli such as heat, light, and chemical attractants or repellents. These are collectively referred to as tactic responses and include chemotaxis [1, 2], thermotaxis [3], phototaxis and rheotaxis [4, 5], among others. A large number of mathematical models have been developed in the study of bacterial chemotaxis, from various angles of approach. The models can be broadly classified into either the agent-based approach [6-10], which takes the single-cell perspective [11], or the population-based approach [12-19], which views bacterial transport from a field perspective. The bacterial field concept has been used in the development of travelling wave solutions describing the collective behaviour of cells [20, 21]. The most general form of the bacterial transport equation is analogous to the equation for molecular transport, with chemotactic drift represented as an advective term that is attractant concentration-driven rather than flow-driven. Tindall et al. [18] collated many mathematical models developed prior to 1990, which include the general transport model formulated by Keller and Segel [12, 22], as well as models of greater complexity that were formulated by Alt [13] and Othmer et al. [14]. More recently, Hillen and Painter [19] conducted an extensive study on Partial Differential Equations (PDEs) for bacterial transport, based on the Keller-Segel model, which focused on the various forms of diffusive, advective and source/sink terms that exist in the literature. In particular, multiple expressions for the chemotactic drift velocity term have been proposed, primarily as functions of local attractant concentration and concentration gradient. The significance of direct versus indirect ligand binding in chemotaxis has previously been discussed [23] but has not been extensively studied. For direct binding, attractant molecules entering the periplasmic space of the cell interact directly with the membrane-spanning chemoreceptor proteins. On the other hand, indirectly-binding attractants interact only with free-roaming binding proteins (BPs) located in the periplasm. The attractant-protein complexes are then able to bind with chemoreceptors and thus trigger the chemotactic response. Indirect binding appears to provide the cell with better control over the chemotactic response by varying the number of available BP molecules [24], with overall response being highly sensitive to expression levels of BPs [23], in exchange for a lowered dynamic range of response. Furthermore, some periplasmic binding proteins appear to serve concurrently in 4

chemosensory and transport roles as components of ATP-binding cassette (ABC) transporters, including the maltose [25] and AI-2 [26] binding proteins. While this clearly allows for coordination in the uptake and chemotactic responses, the specific strategies which result or benefit from this pairing have yet been fully understood. The Rivero chemotactic velocity equation has been successfully used to predict bacterial chemotactic behaviour in a number of studies [17, 27, 28]. The Rivero model is advantageous due to its mathematical simplicity, allowing chemotactic behaviour to easily be incorporated into a general bacterial transport equation as an advective term. However, it assumes that the chemotactic response depends only on a constant chemotaxis coefficient and the two extracellular variable factors of attractant concentration and gradient. The chemotaxis coefficient, in turn, is related to the cell swimming velocity and the total receptor concentration, which are typically assumed constant, as well as an empirical “chemotactic sensitivity constant” [29]. While Brown and Berg [29] refer to this coefficient as an arbitrary time constant, Rivero et al. [15] further describe it as a constant which relates the change in average run duration with the time rate of change of the number of bound receptors. Therefore, it is clear that the existing model does not take the effect of periplasmic binding protein concentration into account. Indeed, the equation specifies a single dissociation constant representing the binding affinity between the chemoreceptor and the attractant ligand directly, which cannot be properly defined for indirect binding mechanics. In the case of the Escherichia coli AI-2 signalling system, the number of periplasmic binding protein LsrB molecules [26] has been reported to fluctuate over time [24]. The LsrB concentration is strongly dependent on the level of expression of the lsr operon [30], which is in turn modulated by the concentration of extracellular AI-2. In the chemotaxis assay experiment, this means that the apparent sensitivity of the sample to the attractant can fluctuate according to the average concentration of LsrB that the cells possess from the beginning of the assay. Furthermore, it appears that the extracellular AI-2 and LsrB concentrations are not always quantified as part of the experimental protocol, meaning that a portion of chemotactic response due to dynamics of self-secreted signalling molecules may be unaccounted for [31, 32]. AI-2 sensitivity profiles may thus vary significantly under apparently similar experimental conditions and attractant source concentrations [24, 26]. An indirect binding model may thus be better suited for characterizing AI-2 chemotaxis by allowing additional parameters such as the periplasmic binding protein concentration to be factored into the chemotactic response. 5

In this study, we first present our formulation of the indirect binding expression as a modification of the existing Rivero chemotaxis equation. We then carry out a series of parametric studies using the chemotactic response of E. coli to maltose and AI-2 as case studies. We first directly examine the indirect binding chemotaxis equation in 1D, followed by simulating capillary assay experiments in 2D and comparing predicted sensitivity profiles with experimental results in the literature. Our findings suggest that physiological levels of both the maltose and AI-2 periplasmic binding protein concentrations are well-adapted for allowing the bacterium to maximise chemotactic response across as wide an attractant concentration range as possible. This observation is made possible by the indirect binding expression which allows binding protein concentration to be expressed as an explicit contributing factor to the variable chemotactic response.

6

2. Methodology 2.1 Preliminaries Rivero et al. [15] formulated a chemotactic velocity expression in 1D for a bacterium that swims with a run-and-tumble motion, such as E. coli. As mentioned previously, the advantage of their chemotactic velocity equation lies in its mathematical simplicity which allows it to be applied in a general bacterial transport model as a self-propelled advection term. The expression, which relates chemotactic drift velocity to the swimming velocity, rate of change of bound chemoreceptors and external attractant concentration gradient, is given by: (

)

(1)

refers to the average swimming velocity of the cell, coefficient,

is the chemotactic sensitivity

refers to the concentration of bound receptors and

is the external attractant

concentration. In their formulation, Rivero et al. utilized the finding of Brown and Berg [29] who determined that the logarithm of the average run time

of a straight swimming cell was

related to the time rate of change of the number of bound receptors: (2) where ⁄ and

is the mean run time in the absence of attractant, and

is a numerical constant,

refers to the total number of receptors. Applying the law of mass action,

can also be related by [33]: (3)

which states that at equilibrium, the fraction of bound receptor is a function of the external attractant concentration

and a dissociation constant

. Thus, this formulation was derived

from the perspective of directly-binding attractants such as amino acids. This form assumes that the substrate or ligand binds directly with the chemoreceptor, such as serine with the Tsr receptor or aspartate with the Tar receptor [34, 35]. It has been found that other chemoattractants interact with a periplasmic binding protein (BP) to form a complex which then binds to the chemoreceptor [36]. Examples include maltose, which binds

7

indirectly to Tar through the MalE binding protein [25, 37, 38], and AI-2, which binds indirectly to Tsr via the LsrB binding protein [26]. We believe that the binding kinetics for indirectly-binding attractants bear similarity to that described by Kehres [39] for the binding protein-mediated arabinose transport system of E. coli. In this work, it is stated that the translocation rate of arabinose into the cell obeys the Michaelis-Menten equation, whereby [

] [

(4)

]

with translocation rate

, maximum reaction rate

and a characteristic Michaelis constant

, ligand-protein bound complex [

. From the law of mass action, [

] is expressed as

a function of the free ligand concentration [ ], total binding protein concentration equilibrium dissociation constant [

]

]

, and the

for the ligand-binding protein pair:

[ ]

(5)

[ ]

Substituting (5) back into (4), an expression relating translocation rate to free ligand concentration is obtained: [ ] [ ]

(6)

[ ]

The main feature of this expression is the explicit dependence on the periplasmic binding protein concentration

on the arabinose uptake rate. We seek to obtain an expression for

the fraction of bound chemoreceptors that exhibits similar dependence on the periplasmic binding protein concentration. 2.2 Formulation of indirect binding expression The enzyme reactions for indirect binding chemotactic sensing are given by the following equation, based on reversible enzyme kinetics [40]:

[ ]: Ligand, [ ]: Periplasmic Binding Protein, [ ]: Receptor [

]: Ligand-BP complex, [

]: Ligand-BP-Receptor complex 8

: characteristic association rate,

: characteristic dissociation rate

Segel formulated a more detailed model of chemotaxis enzyme kinetics [41] by considering the enzyme to exist in two conformational states. Using a simplified approach without consideration of the conformational state of proteins, we formulate a similar but reduced set of equations. By simply considering the mass balance for each chemical species without synthesis or decay, a system of 5 PDEs is obtained: [ ]

[ ][ ]

[

]

(7a)

[ ]

[ ][ ]

[

]

(7b)

[

]

[ ][ ]

[ ]

[

]

[

]

[

][ ]

[

]

(7c)

[

][ ]

[

]

(7d)

[

][ ]

[

]

(7e)

In enzyme kinetics, intermediate chemical species such as [

] and [

] typically begin at

zero concentration, rapidly increase at the start of the reaction, and remain approximately constant until ligand concentration begins to fall significantly near the end of the reaction. The steady-state assumption is applied such that the concentration of ligand-protein complex can be expressed as a function of free ligand concentration and a characteristic binding or dissociation constant [40]. In bacterial chemotaxis, it is not necessary for attractant molecules to be transported into the cell [42]. As such, we assume in our model that the ligand concentration is perfectly conserved, under the condition that the characteristic timescale for observation of chemotactic response is shorter than the timescale for decay. At steady state, [

]

[

[

]

and from Equation 7e,

][ ], where

⁄

Since the total receptor density [

]

[

](

[

is the sum of bound ([

]) for [ ]

[

]) and unbound receptors [ ],

]

9

and thus [ [

] can be expressed as: [

]

Similarly, for [

] [

[

.

(8)

] ]

at steady state, and defining [ ]

[

], we obtain

[ ]

]

(9)

[ ]

where

is the total concentration of periplasmic BP molecules, while

constant for the Receptor-binding protein pair and

is the dissociation

is the dissociation constant for the

binding Protein-attractant pair. Substituting (9) into (8), we obtain: [

[ ]

]

[ ]

(10)

[ ]

From this point, free ligand concentration [ ] will be expressed as the external attractant concentration , and [ (

] expressed as the concentration of bound receptors

.

)

(11)

We assume that the receptor and binding protein concentrations

and

functions of the external attractant concentration . Taking derivative of

do not vary as with respect to ,

we obtain: (

)

(

(12)

)

In the case of attractants that bind to a periplasmic binding protein before binding as a complex to the chemoreceptor, each of the two binding events is characterized by a dissociation constant. In our notation,

refers to the dissociation constant

binding between the attractant and the periplasmic binding protein. Similarly,

of the refers to

the dissociation constant of the binding between the periplasmic binding protein and the chemoreceptor. Substituting Equation 12 back into Equation 1, and grouping

, we obtain in 1D:

10

(

(

)

)

(13)

Brosilow et al. [27] found that the 1D Rivero chemotaxis equation (Eq. 1) could be applied in two dimensions. This approximation was found to agree well with predictions made by the three-dimensional model of Alt et al. [13]. We therefore obtain: (

(

(

)

(

)

The chemotaxis coefficient velocity

)

(14a)

)

(14b)

is formed from the chosen group

to eliminate swimming

from within the hyperbolic tangent function, in contrast with previous studies

which define

[15, 28], sharing the same units as the diffusion coefficient.

While the periplasmic BP for maltose chemotaxis is assumed constant at approximately concentration [43], the periplasmic BP for AI-2 chemotaxis is known to be highly variable [24]. As mentioned earlier, the LsrB concentration is dependent on the level of induction of the lsr operon, which in turn is modulated by the intracellular AI-2 concentration [30]. Ultimately, the LsrB concentration is related to the extracellular AI-2 concentration through a series of reactions, we assume that it remains constant in our current study. 2.3 Capillary assay simulation Capillary assay numerical simulations were conducted using COMSOL Multiphysics® software to model bacterial transport within the 2D geometry of a capillary tube and an external reservoir. Dimensions are shown in Figure 1. The capillary tube is positioned such that the tube entrance is situated at the centre of the reservoir, with no-flux boundary conditions applying to all other interfaces. The free triangular meshing scheme is applied across the whole geometry, with maximum element size of element size of while

and minimum

. Initially, the bacterial density in the reservoir is set at

inside the capillary region. Conversely, we set

,

in the reservoir and

inside the capillary. We assume a constant unit depth (direction into the paper) for both the capillary and the reservoir, with the capillary volume approximated as a cuboid rather than a true cylinder. The relative chemotactic response is reported as the ratio of the

11

number of cells

predicted to enter the capillary against the initial total cell count

cell counts per unit length obtained from the integral of

, with

within each region of interest.

Figure 1: Geometry for simulated capillary assay in 2D (Not drawn to scale). Chemotactic drift velocity is measured along centreline AB along which attractant gradients are steepest. Only the first

from the

entrance of the capillary tube is included according to the reported loading conditions [44].

The predicted cell density within the capillary region is computed by solving the system of two PDEs as given below [12]:

Here,

(

)

(

)

(15a) (

)

(

)

represents the attractant concentration and

the total cell count, defined as the integral of elsewhere by the variable are

(15b) represents the bacterial density, while within a specified region, is denoted

. The values used for diffusion and random motility coefficients

and

[28] throughout the study. The attractant

concentration is assumed to undergo only pure diffusion with zero advection, which is the case for the stationary fluid medium within the capillary and reservoir regions. Effects of decay or depletion of attractant due to bacterial uptake are not considered so as to simplify the analysis and better understand the chemotactic response in isolation. The bacterial density is assumed to be subject to diffusion due to random motility as well as advection due to 12

chemotaxis behaviour only. Cell growth and death rates are not considered as they are assumed negligible within the timescale of the capillary assay. The specified tolerance used for the time-dependent study is tolerance of up to

, with the result appearing to converge for a

.

In Comsol, the PDEs for each independent variable are set up using the “General Form PDE (g)” physics interface in 2D, which provides a classical general PDE equation of the form: (16) where variable,

and

are the damping and mass coefficients respectively,

is the conservative flux term and

the flux expression

field, where

and

is the source term. In the general form PDE, (

is unspecified, thus we define

concentration field and

(

̂

is the independent

̂)

(

̂

̂

̂) for the attractant

̂) for the bacterial density

and the diffusion and random motility coefficients respectively.

The simulation geometry is constructed in two distinct domains. The capillary region is generated from the “Draw Rectangle” tool while the reservoir region is constructed using the “Draw Polygon” tool, with the only common boundary being the interface between the capillary mouth and the reservoir. The presence of an impermeable capillary wall is simulated by leaving a gap of 100 microns between the rectangular boundary of the capillary and the region bounded by the reservoir. Thus, the no-flux boundary condition is automatically applied to all edges except the interface between capillary and reservoir domains. Cell count in the capillary domain is evaluated at the end of the simulation by using the “Surface Integration” function under “Results, Derived Values” and computing the integral of bacterial density

over the entire domain.

2.4 Parameters for indirect binding chemotaxis of E. coli towards maltose The widely accepted concentration of maltose binding protein in the periplasm of E. coli is approximately

[38] based on osmotic shock experimental data and derived from the

average number of molecules per cell. Dietzel et al. [45] reported a count of 20000 molecules per cell, while Kellerman and Szmelcman [46] reported 30000 molecules per cell and Koman et al. [47] reported 45000 molecules per cell. These studies were conducted between 1974

13

and 1979 and there do not appear to be more recent reports quantifying the periplasmic binding molecule concentration to date. Zhang et al. [43, 48] conducted studies using E. coli strains YZ9, YZ11 and YZ12, corresponding to maltose binding protein concentrations of 100%, 23% and 11% respectively, relative to the wild-type E. coli RP437 specimen. Conversely, Manson and Boos [38] utilized a malE+ phenotype with reported

of maltose binding protein per

cells. For a molecular weight of 40661 daltons [38], this is equivalent to roughly . E. coli has a typical cell volume of

and the periplasmic space has

been estimated to be 30% [49] of the total cell volume or smaller than 10% [50]. Ultimately, this works out to a periplasmic maltose binding protein concentration of between

to

for the malE+ phenotype. The

value of

is given in Manson and Boos [38] for binding between maltose

binding protein and the Tar chemoreceptor. The dissociation constant was estimated by analysing the change in chemotactic response of cells in a capillary assay possessing different known levels of maltose binding protein concentration. Subsequent studies generally refer directly to this first reported value of the dissociation constant and there do not appear to be differing reports of the The value of

value in more recent literature.

has been reported to be between

to

.

Szmelcman and Schwartz [51] found the dissociation constant between maltose and maltose binding protein to be report a

. Using an osmotic shock technique, Duplay et al. [52, 53]

value of

intermediate value of between

. Finally, Manson and Boos [38] employed an in their calculations based on an overall range

and

The chemotaxis coefficient

discussed in the work of Schwartz et al. [54]. introduced by Ford et al. [55] and Rivero et al. [15] comprises

the chemotactic sensitivity coefficient and is defined as swimming velocity in 1D,

, where

is the cell

is the total concentration of chemoreceptors and

(See

Equation 2). The average chemotaxis coefficient in the study of Ford et al. was reported as for response of E. coli to fucose, which is a non-metabolizable analog of galactose that binds to the Trg receptor [ref Manson bacterial signal transduction]. More recently, Ahmed et al. reported values of

between

and

in microfluidic experiments involving E. coli and α-methylaspartate [56]. An 14

earlier work [28] involving the same bacteria and attractant reported a slightly higher value of . Finally, Lewus and Ford [57] collated values of

reported in the

literature for responses to fucose, methylaspartate and serine. Fucose responses were consistently in the order of

, while methylaspartate responses were between

and

. Interestingly, the response of Salmonella

typhimurium towards serine was also reported as

. We note that

methylaspartate and maltose chemotactic responses are mediated by the Tar receptor while those of serine and AI-2 are mediated by the Tsr receptor. In our study, the chemotaxis coefficient

is defined as

an average cell swimming velocity of is

and therefore

, the largest value of

. For

based on

while the smallest based on

is

. The simulations presented in the following section are repeated for ,

and

. Comparisons of predicted cell count within

capillaries against existing experimental data, along with evaluation of the maximum predicted chemotactic velocity, clearly identify

as the most suitable value of

the chemotaxis coefficient to be employed in the study. This finding is further elaborated in the results where the effect of varying presented are based on

is discussed in particular, whereas all other results as the default value for the chemotaxis coefficient.

Parameter values used for model validation are reported in Table 1 below. 2.5 Parameters for indirect binding chemotaxis towards AI-2 The dissociation constant LsrB has been reported as

for binding between AI-2 and its periplasmic binding protein by Zhu and Pei [58] and more recently as

by Torcato et al. [59]. One possible reason given for the large difference in measurements is the presence of boron in the study of Zhu and Pei, which binds with the AI-2 molecule and forms a molecule that is not sensed by E. coli, thus explaining the lowered sensitivity. The periplasmic binding protein concentration [24], which reports a value of

has been quantified by one study thus far

molecules of LsrB per cell when chemotaxis is most

active. This corresponds to a concentration of approximately

per cell, or

somewhat larger if the volume of the periplasm is taken into account as well. The same study

15

also quantified levels of periplasmic binding protein for lsr mutants, which varies between 0.2 to 7.5 times of the value given above. The dissociation constant

for binding between the LsrB-AI-2 complex and the Tsr

chemoreceptor has not been quantified in the existing literature. Using the method described by Manson and Boos [38] for quantifying the binding affinity of loaded maltose binding protein with the Tar receptor, we first attempt to estimate the

of loaded LsrB with the Tsr

receptor. The data used was adapted from the work of Jani et al. [24] who tested chemotactic response of various lsr mutants of E. coli to AI-2 using the capillary assay. In their report, they quantified LsrB levels of each mutant, ranging from concentration in the ∆lsrK mutant to approximately

of wild-type LsrB

in the case for the ∆lsrR mutant. A

total of seven data points were extrapolated from their findings, corresponding to mutants with different LsrB levels, and curve fitting was conducted to visualize change in cell count with varying LsrB concentration (Fig. 2). We estimate the half-maximal response to occur at ⁄

, rounding to

for

.

Figure 2: Capillary assay data points extracted from the study of Jani et al. [24]. Capillary cell count is plotted against periplasmic binding protein concentration ratio and

is estimated from the half maximal response.

The maximal response is defined here as the largest value of each plotted curve, corresponding to the rightmost edge. The

value is then found by finding the value of

that matches the half-maximal response from the

equation of the fitted curve. Each line type corresponds to an attractant concentration and set of experimental

16

data points, while each data point within the same series represents an E. coli mutant with a different LsrB concentration.

Finally, we assume that the chemotaxis coefficient magnitude in the case of AI-2, such that chemotactic sensitivity coefficient receptors

should be of the same order of .

is formed by grouping the

, the swimming velocity

and the total number of

. E. coli possesses roughly twice as many Tar as Tsr receptors [24], implying that

the value of

is twice as large for maltose as it is for AI-2 if

and

can be assumed

constant. The average swimming speed of the bacterium typically lies between

to

and can be assumed constant within this range. However, the chemotactic sensitivity describes the change in mean run time per unit time rate of change of receptor occupancy, which condenses the mechanisms of receptor binding, flagellar motor signalling, tumbling inhibition and adaptation into a single empirical coefficient. To the best of our knowledge, the value of

has only been experimentally determined in the original study of

Brown and Berg [29]. Ford developed an analytical expression to estimate the the initial attractant concentration

based on

in a capillary assay, thus highlighting the fact that

,

and by extension , is likely to vary across the chemotactic response range. By evaluating the accumulated cell fraction and maximum chemotactic velocity at different levels of , we find the rationale for choosing

in the case of maltose to apply to AI-2 as well.

Table 1: Parameter values used in study Attractant species Parameter Binding protein concentration

Maltose

AI-2

[38, 45]

Dissociation constant (ligand-binding protein)

[38]

Dissociation constant (binding protein-receptor)

[38]

[24] [58] [59]

Chemotaxis coefficient

17

3. Results and Discussion 3.1 Prediction of chemotactic response to maltose 3.1.1 Scaling of dissociation constant We first conducted a parametric sweep for the dissociation constant

ranging from sub-

micromolar to molar attractant concentrations. For each attractant concentration

, we ran

the capillary assay simulation as described in methods (Fig. 1) and computed the number of cells within the capillary region after a period of one hour. Comparing our results against the experimental work in the literature, we observe that the peak chemotactic response predicted by our model for the physiological

value of

occurs at significantly lower

attractant concentrations than previous experimental reports. We find that

(Fig. 3A) provides a much better fit for the chemotaxis

simulation as compared to the experimental reported value of

over a

wide range of maltose source concentrations (Fig. 3B), based on comparisons with the experimental results presented by Hazelbauer [37], Manson and Boos [38], and Zhang et al. [43]. Notably, this includes E. coli mutants with altered levels of maltose binding protein, ranging from 23% (strain YZ11) to 1000% (estimated value for the malE+ phenotype). In the case of the capillary assay study reported by Hazelbauer [37], we additionally found good quantitative agreement in the predicted cell count for the attractant concentration eliciting peak chemotactic response. In short, the total initial cell count in their study was estimated to lie between

and

, while approximately

cells were recorded in the

capillary at peak accumulation. Using the higher estimation of total cell count, the ratio of cells found within the capillary is thus 0.026, which is matched near identically by our predicted value for

(Fig. 3A, dashed line). Comparisons with other chemotaxis

assay studies yielded qualitative agreement, owing to uncertainty in estimates of the fraction of cells accumulated within the capillary tube. We note, in particular, that our model predicts a distinct shift in peak response from protein concentration is raised to

to

as the periplasmic binding

(Fig. 3A, dotted line) and beyond, which is in

strong agreement with the results obtained by Manson and Boos [38] for malE+ cells. Conversely, when

is lowered to

, corresponding to strain YZ11 [43], the peak

response remains at the same attractant concentration level. Again, this is in agreement with

18

the experimental report, although we are not able to quantitatively determine if the decrease in response from wild-type levels is represented accurately. When the experimentally reported value of

is used in the capillary assay

simulation (Fig. 3B), we observe that the predicted response differs significantly from existing experimental data. Considering specifically the case where

(Figs. 3A &

3B, dashed lines), the peak predicted chemotactic response occurs at two orders of magnitude of attractant concentration

lower than the experimental profile. Furthermore, the predicted

response appears highly saturated at

such that only a minimal number of cells are

able to enter the capillary. This contrasts with numerous experimental reports [37, 60] which indicate maximum response to an initial attractant concentration of

.

Figure 3: Sensitivity profiles based on predicted cell count from capillary assay simulations. The ratio of the number of cells within the capillary region

over the total initial cell count

attractant concentration. Parameter values used are (A): and (B):

,

and , the value of

,

is plotted against initial and

. As the

value reported in the literature

is

or

used in set (A) is thus scaled by a factor of approximately

30.

corresponds to the periplasmic binding protein concentration for wild-type E. coli. corresponds to strain YZ11 from Zhang et al. [43].

to

represent estimated

values of the heightened maltose binding protein concentration in the malE+ strain from the work of Manson and Boos [38].

We believe that the observed difference in effective

value from experimental reports

could be attributed to a difference in the concentration of maltose between the periplasmic space and extracellular medium. Equation 13 relates the extracellular attractant concentration 19

directly to the chemotactic response, making the implicit assumption that the extracellular attractant concentration is equal to the concentration of attractant that has diffused into the periplasm through porins. In general, the extracellular attractant concentration is not the same as the concentration within the periplasm unless the extracellular concentration does not change with time. Even so, active maltose transport from the periplasm into the cell suggests that attractant concentrations within the periplasm remain lower than extracellular levels. Detailed single-cell models of substrate transport [61] track the periplasmic attractant concentration separately from both intra- and extracellular attractant concentrations, while relating them through the rate of diffusion via porins and uptake through membrane-spanning transporters. It suffices in our current study to assume that the difference in attractant concentration across the outer membrane can be satisfactorily addressed by scaling the parameter

. For the experimentally reported value

, the scaling factor

thus appears to be approximately 30. Based on this finding, we introduce the scaling factor (

(

to Equation 13 such that:

)

)

The change in chemotactic response with increasing values of

(17a) are plotted in Figure 4. The

response profile translates by one order of magnitude to the right for each tenfold increase in the value of

, showing that

concentration

scales proportionally with the extracellular attractant

. Indeed, the above equation can be rearranged to express that the

extracellular attractant concentration scales inversely with : (

(

)

).

(17b)

Equation 17b indicates that the extracellular attractant concentration is effectively reduced by a factor of

in the periplasm. This simplification of the actual dynamic fluctuation of

attractant concentration in the periplasm may be justified in conditions where the time rate of change of periplasmic attractant concentration can be assumed constant.

20

Figure 4: Change in predicted chemotactic response with varying

. The sensitivity profile undergoes pure

translation towards the right for each tenfold increase in the value of . Peak chemotactic response shifts rapidly from

to

between

and

and reaches a maximum value of approximately

0.025 (ratio of cells accumulated in capillary against total number of cells) by

. Subsequent increase in

has a more pronounced effect on the chemotactic response to the next highest attractant concentration level, in this case where

.

The curve of

(Fig. 4) corresponds to Figure 3B where all parameters are unchanged

from their experimentally reported values, while the curve of

corresponds most

closely to the estimated best-fitting case in Figure 3A with

. The chemotactic

response to

predicted by

is near maximal and further increases in

in more pronounced response toward the initial attractant concentration of When

exceeds 50, the response to

result .

becomes greater than the response to

. Since the peak chemotactic response remains at

between

to

, the relative strength of predicted responses to the adjacent attractant concentrations of C0 may be useful in refining the estimation of . Presently, the value of

has been

chosen to best represent the existing experimental data [37]. As shown in Figure 3A with ,

predicts a relatively larger response at

than

,

matching the earlier experimental report of Hazelbauer. 3.1.2 Effect of other parameters on predicted chemotactic response We considered the possibility that the difference between predicted and actual chemotactic sensitivity curves (Fig. 3B) could be attributed to errors in the value of the other parameters 21

and

. First, we note a key difference in the methods in which

experimentally quantified. The dissociation constant

and

were

, for maltose with maltose binding

protein, reported in previous literature was measured through experiments carried out using maltose binding protein that was released from the E. coli cell periplasm via osmotic shock. As per the earlier discussion, this implies that the experimentally determined value of must be applied together with the actual attractant concentration of the space within which the binding occurs, i.e. the periplasmic attractant concentration. On the other hand,

was

quantified by Manson and Boos [38] based on chemotaxis capillary assay data using strains of E. coli possessing varying levels of maltose binding protein. By plotting chemotactic response against binding protein concentration, they were able to estimate the concentration of binding protein which corresponded to the half maximal response defining the dissociation constant

. Thus, we believe that the reported value of

is unaffected by differences in

extracellular and periplasmic attractant concentrations as it was quantified based on the cellular response at the population scale. To test this, we vary

from

to

while keeping all other parameters at their

default levels. We find that the change in response differs (Fig. 5A) depending on whether is raised or lowered from the baseline value of studies. Lowering

reported in experimental

results in a shift in the sensitivity profile towards smaller values of

within the range of

tested, which is a similar effect to that achieved by lowering

.

Conversely, raising

does not cause the sensitivity profile to shift toward higher attractant

concentrations any further. Instead, we observe that the overall chemotactic response diminishes as

is increased past

, indicating that it is not possible to achieve good

agreement with the experimental sensitivity profile by manipulating the chemotactic response at changes in the value of

alone. Furthermore,

and above is almost completely insensitive to

.

22

Figure 5: (A) Change in predicted chemotactic response with varying is

(dotted line). Lowering

causes the sensitivity profile to shift towards more dilute attractant

concentrations (solid and dashed lines), while increasing in the total chemotactic response to all levels of limit to the sensitivity range ( originally estimated value of value of

. The physiological or baseline level of

above the baseline value results in a net decrease

. Changes in the level of

do not appear to affect the upper

and above). (B) Change in chemotactic response with varying as applied to all other simulations is

. The

. While increases in the

result in a shift in the peak response to higher levels of attractant concentration as well as a greater

range of response, this is accompanied by a large increase in the fraction of cells accumulating in the capillary beyond physiologically observed levels. There is negligible response for values of

By quantifying

smaller than

.

from the total concentration of periplasmic binding protein rather than the

fraction that is bound to maltose, it is possible that the actual sensitivity of binding between receptor and maltose-bound binding protein is underestimated. Thus, we expect the value of to be smaller if it were to be quantified using the osmotic shock or similar method. As this would cause the discrepancy between the predicted and experimental sensitivity profiles to become larger (Fig. 5A, solid and dashed lines), it appears highly unlikely that an error in the value of

is responsible for the difference between experimental and predicted

chemotactic responses. Increasing the chemotaxis coefficient

by orders of magnitude causes the sensitivity profile

to shift to the right and the range of response to widen, while also greatly increasing the ratio of cells accumulated in the capillary (Fig. 5B). Conversely, the fraction of cells entering the capillary becomes negligible when

is lowered to

, the peak chemotactic response is shifted to

and below. In the case of , thus agreeing with

previous experimental results. However, the predicted fraction of cells is large in comparison 23

with experimental data which report the cells accumulated in the capillary to be 10% of the estimated total number of cells based on initial cell density or less. Moreover, the predicted chemotactic velocity is extremely high (

) for responses to all tested attractant

concentrations, representing the unlikely scenario where nearly all cells are swimming deterministically in the direction of the steepest attractant concentration gradient. It is apparent that scaling the chemotaxis coefficient has a simultaneous effect on the fraction of cell accumulation, average chemotactic velocity and peak chemotactic response, making it impossible to address the difference between experimental and simulated chemotactic response simply by adjusting

alone.

We also compare the maximum predicted chemotactic velocity to of

for each value

(Fig. 6) across the entire simulation time range of one hour. The chemotactic velocity is

represented as the maximum value of

along line AB (Fig. 1), which runs along the

capillary axis across the full length of the capillary and reservoir combined. In general, the maximum chemotactic velocity occurs at the entrance of the capillary where attractant gradients are steepest. As mentioned in the previous paragraph, the predicted chemotactic velocity for

is improbably large (Fig. 6, solid line), while falling far below

of swimming speed at the case of

and below. The maximum velocity predicted in (Fig. 6, dashed line) remains large (

) for the first twenty

minutes or so. Noting that the peak chemotactic response occurs at

instead of

(Fig. 5B, dashed line), we expect the maximum predicted chemotactic velocity to be somewhat higher at the attractant concentration eliciting the peak response. Consequently, it appears that simulations run using

may predict an excessively large

chemotactic velocity for a certain range of the attractant concentration

.

24

Figure 6: Ratio of maximum chemotactic velocity (See Fig. 1) for different levels

against swimming velocity

evaluated along line AB

over a simulation time of 1 hour. The typical ratio of chemotactic velocity to

swimming speed is approximately

[2], although higher values such as

[28]. The chemotactic velocity ratio predicted by

have been reported as well

is thus excessively high and does not match

observations based on run-and-tumble swimming behaviour. The predicted chemotactic velocity always begins at the maximum value due to the discontinuous jump in attractant concentration across the mouth of the capillary at the beginning of simulation. Disregarding early time-steps, the chemotactic velocity ratio predicted by

persists above when

for approximately 20 minutes. Conversely, the ratio rapidly falls below

and remains at roughly

at the end of simulation time.

3.2 Prediction of chemotactic response to AI-2 The two values of

reported in the literature for binding of AI-2 to the LsrB periplasmic

binding protein are

and

response based on the product

. We first study the chemotactic

in lieu of evaluating the suitability of each possible

value. The change in chemotactic response for different values of to

ranging between

are plotted in Figure 7. As before, the chemotactic response profile

undergoes pure translation to the right for each order of magnitude increase in

(Fig.

7A). Narrowing the range of observation (Fig. 7B), we find that setting

gives

a simulated response that matches the experimental observations of Hegde et al. most closely. In their study, the cell accumulation was reported in terms of the estimated actual cell count, which we approximated as a ratio of the total cell number using their reported initial cell density of

and assuming a reservoir volume of

. This volume is

consistent with the dimensions described by Adler in the original capillary assay study [44].

25

The value of

corresponds to

and

for the higher value of

for

. We believe that the latter value of

is

less likely as it implies that the periplasmic concentration of AI-2 may be as small as while the extracellular concentration is in the order of

. The passive rate of

diffusion of AI-2 into the cell periplasm via porins is unlikely to differ greatly from that of maltose molecules. Thus, the main factor governing periplasmic concentrations of attractants appears to be the specific rate of uptake via transport apparatus. Present information is insufficient to conclude that value of

is indeed overly large, and by extension that one

is more suitable for characterising AI-2 chemotaxis than the other.

Figure 7: Predicted chemotactic response at different values of

: (A)

sensitivity profile undergoes pure translation in the direction of increasing increase in

. (B)

to

to

. The

for each order-of-magnitude

. The curve predicted by setting

corresponds

most closely to existing experimental reports, which indicate that the chemotactic response to AI-2 increases exponentially from

to

(

). The experimental data curve is

estimated from the capillary assay data in Hegde et al. [26] using their reported cell density of and assuming the volume of the Adler capillary assay to be

.

The concentration of periplasmic binding protein in the AI-2 system of E. coli is known to vary over time. The experimental report of Jani et al. [24] suggests that the level of binding protein LsrB varies between 0.2 to 8 times of the characteristic concentration of approximately

. The low level of LsrB corresponds to the lsrK mutant which

only expresses basal levels of the Lsr proteins and cannot be further induced due to the deficiency in production of phospho-AI-2. The high level of LsrB corresponds to the lsrR 26

mutant which is assumed to express maximal levels of proteins encoded by the lsr operon as a result of complete de-repression. We examine how the level of periplasmic binding protein affects chemotactic response of E. coli to AI-2 across the range of LsrB discussed above. Our model predictions do not agree very closely with the experimental results of Jani et al. [24] for the chemotactic response of lsr mutants (Fig. 8). Using

to represent the dissociation constant of AI-2 and

LsrB binding, we compare the chemotactic response at three different levels of binding protein concentration

. Their experimental findings convey a trend that the relative

increase in cell count per level of attractant concentration

decreases steadily and gradually

approaches a maximum cell count over the range

to

.

Conversely, our model predicts that the number of cells entering the capillary region continues to rise for each increment in the level of

over the same range, which is in closer

agreement with the earlier study of Hegde et al. (Fig. 7B).

Figure 8: Predicted chemotactic response to three different levels of LsrB periplasmic binding protein concentration with

,

and

.

corresponds to the

lsrK mutant which remains at baseline levels of lsr induction due to its inability to produce phospho-AI-2. approximates the lsrR mutant which is in a fully induced state as the lsr operon is never repressed. The chemotactic response at

is relatively large at

as compared to the other curves. The

cell fraction is estimated from data of Jani et al. [24] based on an initial cell density of

.

3.3 Application of the indirect binding chemotaxis model

27

Thus far, we have performed validation for the formulated indirect binding model by simulating the chemotactic response of the model organism E. coli towards the attractants maltose and AI-2. Maltose binding protein is encoded by the malE gene and binds to the major chemoreceptor Tar, while LsrB binds AI-2 with the major chemoreceptor Tsr. In general, the model predicts the maltose chemotactic response with good agreement with existing experimental reports. Conversely, the model does not achieve as high a level of agreement for the predicted AI-2 chemotactic response, noting that the validation is limited by the availability of experimental data for AI-2 chemotaxis. We envision that our existing model can be refined in two areas in order to better predict the AI-2 chemotactic response. Firstly, in our formulation, we defined

[ ]

[

], which states that the total

periplasmic binding protein concentration is the sum of the free concentration and the ligandbound concentration. This simplification assumes that the fraction of [ receptor [ ], forming the complex [

] bound with

], is negligible due to the relatively small number of

receptors present in total. While this has been noted to be generally true for indirect chemoreceptor systems [24], the concentration of periplasmic binding protein LsrB is significantly lower than the total receptor concentration in the case of AI-2 [24]. Thus, it may be more appropriate to modify our formulation by defining

[ ]

[

]

[

] to

specifically model AI-2 chemotaxis in a future work. Presently, our aim is to present a general model for indirect binding chemotaxis based on the prevailing notion that periplasmic binding protein is typically in abundance as compared to receptor molecules. Secondly, an indirect binding model for AI-2 chemotaxis may require additional consideration for the variable periplasmic binding protein concentration. The time-scale for significant ramp-up of the LsrB concentration appears to be in the order of hours, as observed from the onset of AI-2 depletion in an E. coli batch culture after four hours had elapsed [31] (LsrB also participates in the uptake of extracellular AI-2). Therefore, it remains unclear if the level of LsrB varies significantly over the course of a typical capillary assay lasting between 45 minutes to an hour. Single-cell models linking intra- and extracellular AI-2 levels with expression levels of lsr operon-mediated proteins [7, 61] may be partially employed in conjunction with an indirect binding chemotaxis model to simulate changes in the periplasmic binding protein concentration over time. In addition to the two factors discussed, the inclusion of AI-2 uptake mechanics may alter the extracellular AI-2 concentration profile sufficiently to cause a significant change in the predicted chemotactic response.

28

Other indirectly binding ligands identified for E. coli include ribose, galactose [62] and dipeptides [63, 64], which interact with the minor receptors Trg and Tap. Although Salmonella typhimurium shares four chemoreceptors that are orthologs of those found in E. coli [65], it is unclear if this similarity extends to the employment of periplasmic binding proteins in chemotactic responses to the corresponding ligands. Indirectly binding ligands have also been observed to exist in a number of other species of bacteria [65], although they do not appear to have been extensively studied. In Bacillus subtilis, the McpC receptor appears to utilize an indirect binding mechanism in the chemotactic response to four amino acids, which are lysine, arginine, methionine and glutamine [66]. Thus far, periplasmic binding proteins associated with these ligands have yet to be identified. With multiple periplasmic binding proteins potentially interacting with the same receptor, it may be necessary for a more robust indirect binding chemotaxis model to account for competition between different attractants. The pathogen Helicobacter pylori responds to AI-2 as a chemorepellent [67] through an indirect binding chemotactic response mediated by two separate periplasmic binding proteins that appear to function independently [68]. The Rivero chemotaxis model [15] that forms the basis of our indirect binding model is formulated to include the possibility of both chemoattraction and chemorepulsion. However, examples of the base model being applied in the prediction of chemorepulsive behaviour are scarce. Thus, the challenges anticipated in applying the indirect binding model to predict the chemotactic response of H. pylori with AI2 include: Validation of the model as well as the underlying Rivero model for chemorepulsion in general, and extension of the indirect binding model to include two independent periplasmic binding proteins that may function cooperatively or competitively. Finally, Pseudomonas aeruginosa has been found to bind inorganic phosphate (Pi) through two separate chemoreceptors, of which one binds to Pi directly and the other through a mediating periplasmic binding protein [69]. In this work, it is noted that the identified binding protein, PstS, is the most abundant protein in P. aeruginosa when grown under conditions of scarce inorganic phosphate. This observation recalls a similarity with the E. coli maltose chemotactic response in which binding protein concentration is significantly higher than the total receptor concentration, such that the assumption

[ ]

[

] is valid. Nevertheless,

modelling the chemotactic response of P. aeruginosa to inorganic phosphate appears to require both a direct and indirect binding chemotactic expression. In this case, it remains

29

unclear if their effects on the chemotactic velocity are merely additive or related through a higher-order expression. The limited examples of indirect binding chemotactic response mechanisms in various bacterial species presented here serve to highlight that each chemotactic response system may differ significantly from one another. Therefore, we suggest that the indirect binding model presented in the current work may serve as a general or base model, to which further refinements can be made to specifically cater to the unique characteristics of a given chemotactic response system.

Conclusion In this study, we have formulated an indirect binding expression which builds upon the original chemotaxis model of Rivero et al. The indirect binding expression takes the binding of ligand to periplasmic binding protein into account, followed by binding of the ligandperiplasmic binding protein complex to the chemoreceptor itself. The kinetic equilibrium state of the indirect binding thus involves two dissociation constants and is dependent on the periplasmic binding protein concentration, such that the predicted chemotactic response is non-existent in the limit that the periplasmic binding protein concentration is zero. We additionally find that a clear distinction must be made between extracellular and periplasmic attractant concentration in our model. By assuming that attractant concentrations in the two spaces are equal, our model appears to significantly over-estimate the sensitivity of the chemotactic response such that the predicted peak response occurs at much smaller attractant concentrations than the experimental data. We address this matter by introducing a scaling factor representing the possible difference in attractant concentration levels between the extracellular and periplasmic spaces. Alternatively, the scaling factor can be interpreted as an effective lowering of the overall sensitivity. The formulated expression is tested by conducting capillary assay simulations for the chemotactic responses of the E. coli bacterium to maltose and AI-2. Using values for the dissociation constants and periplasmic binding protein concentration reported in the previous literature, we find that the maltose response, in particular, is in close agreement with the model predictions using a scaling factor of approximately 30. On the other hand, our model is consistent with one experimental study on AI-2 chemotaxis, out of two surveyed in the 30

existing literature. We believe that the accuracy of the model may be affected by two characteristics of the AI-2 chemotactic response system of E. coli. Firstly, the periplasmic binding protein concentration is much lower than the cognate receptor concentration. Thus the fraction of receptor-bound binding protein may be significant, contradicting with our underlying assumption that it can be fully neglected. Secondly, the variable level of periplasmic binding protein LsrB may result in significant changes in the magnitude of chemotactic response over time. Furthermore, as LsrB is also implicated in the AI-2 uptake mechanism, it is possible that the attractant concentration profile is significantly altered after a sufficient period of time. Taking these factors into account, we envision that the current indirect binding model can be further refined to provide deeper insight into the complexities of the chemotactic response of E. coli towards AI-2.

31

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