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Travelling wave analysis in chemotaxis: case of starvation
- P. M. Tchepmo Djomegni^{1}Email author
- Received: 31 March 2016
- Accepted: 2 June 2016
- Published: 29 June 2016
The Erratum to this article has been published in SpringerPlus 2018 5:2121
Abstract
In this paper we investigate the existence of travelling wave solutions for a chemotaxis model under the scenarios of zero growth and constant growth rate. We use Lie symmetry analysis to generate generalized travelling wave solutions, a wider class of solutions than that obtained from the standard ansatz. Unlike previous approaches, we allow for diffusivity and signal degradation. We study the influence of cell growth, diffusivity and signal degradation on the behaviour of the system. We apply realistic boundary conditions to explicitly provide biologically relevant solutions. Our results generalize known results.
Keywords
- Group-invariant solutions
- Travelling wave solutions
- Metabolism
Background
Chemotaxis is orientation (or movement) of an organism in response to chemical signals. Cells, through membrane receptors located at their surface, sense the environment, detect chemicals, and then transfer information to their interior (Berg et al. 2002). Depending on the nature of the information, an enzyme will be produced and will cause cells to respond accordingly (attraction or repulsion). The protein CheY facilitates the transmission of the signal from the chemoreceptors to the flagella motors in E. coli (Paul et al. 2010). The phosphorylation of CheY caused by chemorepellents will drive the flagella to rotate clockwise, and the dephosphorylation of CheY caused by chemoattractants will drive counter-clockwise rotation of the flagella (Maki et al. 2000; Eisenbach and Lengeler 2004). Counter-clockwise rotation of the flagella causes the cell to move forward, and clockwise rotation causes the cell to stumble (we note that E. coli moves by jumping through the rotation of its flagella).
Progress made in cell biology shows that chemotaxis plays a vital role in reproduction, tissue repair, drug delivery and tumor invasion (Entschladen and Zanker 2002; Friedrich and Jülicher 2007; Schneider et al. 2010; Lajkó et al. 2013; Sahari et al. 2014). In fact, sperm cell motility is directed by chemoattractants resulting from signalling of female reproductive tract (Entschladen and Zanker 2002; Friedrich and Jülicher 2007). In wound healing processes, chemotaxis facilitates the aggregation of immune system cells into site of infection (Schneider et al. 2010). It is also involved in metastasis and atherosclesis states of diseases (Condeelis et al. 2001; Devreotes and Janetopoulos 2003; Gangur et al. 2002; Moore 2001; Murphy 2001). In pharmacology, chemotaxis is involved in drug delivery to the targeted defective area (Lajkó et al. 2013; Sahari et al. 2014). The beauty of the dynamics of chemotaxis is that cells manifest harmonious behaviour, while behaving independently. This was observed independently by Engelmann (1881a, b), Pfeffer (1888) and Beyerinck (1895). With the remarkable work of Adler (1966a, b) in the past fifty years, bacterial chemotaxis became one of the better-documented systems in Biology. Adler (1966a, b, 1975) observed travelling bands of bacteria when he introduced a population of cells (E. coli) in a capillary tube accommodating oxygen and an energy source. Two bands of cells were formed; the first band consumed all the oxygen and the second band consumed the residual energy source. Bands were also observed without the adding of the energy source; cells consumed oxygen and excreted a gradient of energy source (Adler 1966b). Bak et al. (1987) noticed that the bands were in a form of a circular ring. The complexity of the geometric patterns caused by chemotaxis cannot be intuitively explained from experiments (Murray 2002). As a result, mathematical modelling approaches have been proposed which have been able to predict the geometric shape of the pattern (Keller and Segel 1970, 1971a, b; Patlak 1953; Scribner et al. 1974; Hillen and Painter 2009).
From the cell-based perspective, Patlak (1953) proposed the first model for chemotaxis to depict the random walk process of a particle with external bias and persistence of direction. This model was later improved by Alt (1980) and Othmer et al. (1988). Recently, Xue et al. (2011) formulated a model which takes into account the interaction between two substrates (nutrients and attractants). What is remarkable about their model is that it can be applied in a variety of biological situations, including population dynamics to describe the competition between two species from a microscopic level (the individual species behaviour). Variables describing intracellular processes such as metabolism and transduction of the signal were explicitly represented in the Xue et al. ’s (2011) model. Travelling wave solutions with a unique wave speed were demonstrated in the scenario of zero growth, without requiring a singularity in the chemotactic sensitivity. We Tchepmo Djomegni and Govinder (2015b, 2016) extended these results by allowing for diffusivity and cell proliferation, and provided explicit solutions for the first time. Franz et al. (2014) studied the case of starvation. They assumed that cells consume chemoattractants only (which do not diffuse over the space), and considered a non constant growth of bacteria. They proved the existence of travelling wave solutions in the case of no chemotaxis. It has been proved that the parabolic limit of the microscopic model is the Keller–Segel model (Lui and Wang 2010).
In this paper, we will be looking at the individual behaviour of cells to understand the convergence and harmonization of their motion. The aggregation and movement (with constant speed) of cells are the centre of our study. We will focus on the case of low presence (or absence) of nutrients as the formation of bands of cells was observed in this situation (Adler 1966b; Brenner et al. 1998). The existence of travelling wave solutions will be investigated. Unlike previous approaches, we will allow for diffusivity, and will account for signal degradation and constant cell growth. We will also study the impact of microscale parameters (such as cell growth rate, cell unbiased turning rate and cell speed) on the macroscopic behaviour of the system.
We introduce the model in “Reduced model and analysis” section. As symmetry analysis has proven to be very effective in finding useful solutions to PDEs (Clarkson 1995), we utilise that approach for our system of PDEs. We generate a class of invariants that lead to generalized travelling wave solutions. In some cases, we utilise dynamical systems analysis to further investigate the behaviour of the solutions. (This confirms our previous findings on the interplays between group theory and dynamical systems analysis Tchepmo Djomegni and Govinder 2014.) Realistic initial and boundary conditions are then applied to obtain relevant solutions. We discuss our results in “Discussion” section.
Reduced model and analysis
The system (11–13) was analysed in the case of no chemotaxis by Franz et al. (2014). They assumed that \(D_{S}=\gamma =0\), and h(S) is a linear function of S. Due to the complexity of the system, diffusivity has always been ignored in the mathematical analysis. It is important to note that diffusivity plays a stabilizing role in the behaviour of the system (Rosen 1977). As a result, in our analysis, we will allow for diffusivity, and will investigate the existence of travelling wave solutions under zero growth and constant growth scenarios [We note that demonstrating the existence of traveling wave solutions is equivalent to demonstrating the existence of solutions to (11–13) (Lui and Wang 2010)]. The impact of the growth rate in the behaviour of the solutions will be explored.
Lie symmetry analysis
Case of zero growth
Proposition 1
For \(-\gamma <c_{2}\le 0\), the function \(S(x,t)=S_{1}(u) {\rm e}^{c_{2}t}\), where \(S_{1}(u)\) is given by ( 49 ), is bounded if and only if \(c_{2}^{1}=c_{2}^{2}=0\).
Proof
We choose \(c_{2}\le 0\) in order to produce damped solutions. We note that the functions \(I_{k_{i}}(v)\) and \(K_{k_{i}}(v)\) given in (49) are continuous and positive (McLachlan 1955; Olver et al. 2010).
Given that the only point in which \(S_{1}(u)\) may not be continuous is zero, and that \(S_{1}(0)\) exists, then the convergence of S(x, t) at the boundaries guarantees the boundedness of S(x, t).
Theorem 1
Proof
Invoking Proposition 1, we note that n(x, t) and S(x, t) are positive, continuous and bounded, and the function \(\hbox{e}^{-(c/(2D_{S}))u}\) is monotonically decreasing. We only need to show that \(S_{1} \in Y_{S}\).
When \(u\ge 0\), the function \(I_{k_{2}}\left( \alpha _{2} \hbox{e}^{-(\sigma _{2}/2) u} \right) \) is decreasing. Therefore, \(S_{1}(u)\) is also monotonically decreasing, and converges to zero as \(u \rightarrow \infty \).
In the case of no chemotaxis (i.e., \(\chi =0\)), we have \(\lambda ^{1}=0\). Then from (45), N(u) blows up as \(u\rightarrow -\infty \); cells can only aggregate in the half plane \(u\ge 0\). Travelling wave solutions satisfying \(S_{1} \in Y_{S}\) do not exist. This is consistent with Xue et al. (2011). However, if we relax the assumption on \(S_{1}\), we can demonstrate travelling wave solutions.
Theorem 2
Case of constant growth \(h(S)=\alpha _{0}\)
No chemotaxis (\(k \rightarrow \infty \))
Theorem 3
The non-diffusing solutions \(S_{1}(u)\) of Theorem 3 are obtained directly by integrating the first order system (59–61). We note that all of the solutions are bounded, for they are continuous and converge at the boundaries. A negative flux (see (66)) simply means that most of the cells move to the left (recall that \(j=s(n^{+}-n^{-})\)). Unlike Franz et al. ’s (2014) results, we do not require a minimal wave speed. This therefore constitutes a generalization of their findings.
Theorem 4
High chemotactic sensitivity (\(k \rightarrow 0\))
Theorem 5
Theorem 6
Discussion
In this paper we studied the existence of travelling wave solutions (with a single peak for the signal) of a microscopic model for chemotaxis. We focused on the case of starvation; cells in this situation consume signal only. The effect of microscale parameters in the stability of the system was examined. Unlike previous approaches, we allowed for degradation of signal (\(\gamma \ne 0\)). While we will compare our results to those previously obtained, it must be borne in mind that this important biological process was not considered in other results. We performed a Lie symmetry analysis to generate a large class of invariants leading to generalized travelling wave solutions. Only relevant invariants were considered, but we believe that rich information could have been extracted from the full form of invariants in different contexts. We provided explicit solutions, many for the first time.
We first considered the case of zero growth. Such a scenario is possible if the time interval is shorter than the period required for cell proliferation. When we imposed no chemotaxis, we could not find travelling wave solutions satisfying \(S_{1} \in Y_{S}\). We note that Xue et al. (2011) also indicated the absence of travelling wave solutions when there is no chemotaxis. However, their results held in the case of no diffusivity (\(D_{S} =0\)). We have shown here that these results also hold in the diffusing case. However, when we relaxed the restriction on \(S_{1}\) (less important in the absence of chemotaxis), we obtained both diffusing and non-diffusing travelling wave solutions, distributed in a half plane (see Fig. 2).
If we now consider the high chemotactic limit, we find that non-diffusing (\(D_{S}=0\)) travelling wave solutions do not exist. This is in contrast to Xue et al. ’s (2011) results in which they were found to exist. The degradation of the signal removed this possibility in our results. However, as evidenced in Theorem 1, the incorporation of diffusion does allow for the existence of travelling wave solutions. It is interesting to note that diffusion, in a sense, counteracts the wave eradication effect of the degradation of the signal.
In order to model the behaviour of cells over more realistic time frames, we incorporated cell growth into our model. As a first attempt we assumed constant growth. In general, with no chemotaxis, non-diffusing (\(D_{S}= 0\)) travelling wave solutions do not exist. However, we observe that this occurs due to the requirement that \(S_{1} \in Y_{S}\). This is not necessary in the case of no chemotaxis. Relaxing this restriction leads to non-diffusing travelling wave solutions (see Theorem 3). Note that, unlike Xue et al. (2011), we dot not require a minimal wave speed. We are also able to find diffusing travelling wave solutions (with a discontinuous flux) provided the growth rate dominates the dynamics (see Theorem 4).
If we consider the high chemotactic limit we find that non-diffusing travelling wave solutions do not exist. However, incorporating diffusivity leads to the possibility of travelling wave solutions (see Theorems 5 and 6 ). Note that these are the first results in the case of high chemotaxis with non-zero growth. In contrast to Keller and Segel’s (1970, 1971a) results in the macroscopic model under zero growth, none of our travelling wave solutions required a singularity in the chemotactic sensitivity.
When cells are highly sensitive to signals, allowing for diffusivity, we observe for \(\alpha _{0}<2\lambda _{0}\) that the total cell population (given by \(T_{2}=2sN(0)/(2\lambda _{0}-\alpha _{0})\)) increases as the growth rate \(\alpha _{0}\) becomes large; here most of the new born cells remain in the band. For \(\alpha _{0}=2\lambda _{0}\), the growth rate controls the behaviour of the system (see § 2.3.2). In this case, the total cell population (given by \(T_{3}=2sN(0)/\alpha _{0}\)) decreases as \(\alpha _{0}\) is large. This is due to the local depletion of the signal; some cells will move towards regions with higher concentrations of signal (this is typical in chemotactic systems). The aggregated cells here do not disperse as we demonstrated the existence of travelling wave solutions in this situation. However, for \(\alpha _{0}>2\lambda _{0}\), we notice instability. The cell growth rate controls the behaviour of the system, and prevents formation of the aggregation.
The inverse phenomenon is observed in the limiting case where cells are not sensitive to the signal gradient; they move randomly in this situation. For \(\alpha _{0} \le 2\lambda _{0}\), we obtained instability in the system, travelling wave solutions do not exist. However, for \(\alpha _{0}>2\lambda _{0}\), the stability of the solutions is controlled by the growth rate \(\alpha _{0}\). We imposed restrictions on the initial conditions in order to foster a collective behaviour. Travelling wave solutions then resulted. This result is in agreement with Lauffenburger et al. ’s (1984) findings (in the macroscopic model), in which travelling wave solutions exist due to the balance of growth, death and random motility.
As result of our investigation, we remark that cell growth and cell unbiased turning rate play an important role in the stability of the system and the aggregation of cells. We also remark that the total cell population in the case of zero growth (\(T_{1}\)) is less than that of the case of constant growth (\(T_{2}\) and \(T_{3}\)). The wider band of cells is obtained in the case of no chemotaxis (we recall that here the growth rate \(\alpha _{0}\) controls the stability of the system, the absence of sensitivity to stimuli keep most of cells in the band). The distribution of cells are displayed in Figs. 1a, 2a, 3a, d, 4a, 5a and 6a. We also note that the total cell population \(T_{i}\) decreases as \(\lambda _{0}\) becomes large; the permanent change of direction does not necessarily destabilize the formation of bands of bacteria.
Conclusion
Table summarizing our findings on the existence of travelling wave solutions (TWS)
h(S) | k | \(D_{S}\) | TWS | Restriction on \(\alpha _{0}\) |
---|---|---|---|---|
0 | \(\infty \) | 0 | \(\times \) (or \(\surd \) if \(S_{1} \notin Y_{S}\)) | – |
\(\ne 0\) | \(\times \) (or \(\surd \) if \(S_{1} \notin Y_{S}\)) | – | ||
0 | 0 | \(\times \) | – | |
\(\ne 0\) | \(\surd \) | – | ||
\(\alpha _{0}\) | \(\infty \) | 0 | \(\times \) (or \(\surd \) if \(S_{1} \notin Y_{S}\)) | – (or arbitrary) |
\(\ne 0\) | \(\surd \) | \(\alpha _{0}>2 \lambda _{0}\) | ||
0 | 0 | \(\times \) | – | |
\(\ne 0\) | \(\surd \) | \(\alpha _{0}\le 2 \lambda _{0}\) |
Notes
Declarations
Acknowledgements
The author would like to thank Prof Kesh for his comments on this paper.
Competing interests
The author declares that there is no competing interests.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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