Analysis of productivity and stability of synthetic microbial communities

Abstract

Bioreactors that employ a synthetic microbial community hold potential to overcome limitations of those based on a single species, which embrace a higher level of complexity due to the inter-species interactions. In this work, a number of generic system structures involving two cross-feeding species and various types of inhibition have been studied, together with two three-species cases where a third species is introduced to fulfil a specific function. These cases are represented by mathematical models and inspected through bifurcation analysis and numerical simulation to reveal how the system structure and parametrization affect stability and productivity of the bioreactor. The results show that inhibitions generally lead to reduction in both productivity and stability, and that the presence of a negative feedback loop and a positive feedback loop may give rise to oscillation and bi-stability, respectively, depending on the strength of the inhibitions involved. The intended gains by the introduction of a third species may be achieved when its negative side-effect is sufficiently moderate, and at the cost of reduced stability. As observed in several cases, the changes in stability and productivity do not always follow the same trend, implying trade-off between the two objectives in the engineering of such bioreactors.

1. Introduction

In recent years, a number of attempts have been made in developing bioprocesses based on artificially formed consortia of multiple microorganisms, referred to as synthetic microbial communities, to gain advantages over the conventional, mono-culture based processes [1–4]. A synthetic microbial community has the potential to complete complex tasks, which are either impossible or inefficient to be carried out by a single wild-type or genetically-engineered species, through making the best use of (i) the properties and capacities of each member of the community and (ii) the interactions between different members [4,5]. On the other hand, such systems inevitably involve a greater level of complexity, originating from the interactions between the multiple species, which requires careful analysis and design in order to achieve the intended engineering objectives [6].

In order to develop a better understanding of the inter-species mechanisms, theoretical analysis based on either analytical models [7] or numerical simulation [8] could play an important role. Existing literature has reported a number of investigations on the interactions of microbial species within natural or artificial communities, particularly focusing on inter-species competition. It is well-known that two microbial species competing for a single rate-limiting substrate are not able to coexist under steady-state operation [9], however, there are specific operational settings that are able to make the coexistence possible [10–13]. In addition to competitive communities, a two-species symbiotic ecosystem was studied, where quorum sensing was shown to impose a significant impact on the stability of the consortium [14]. Previous studies have also investigated ecosystems with more than two species, such as a three-species system competing for two resources [15]. Further, several studies have analysed syntrophic consortia where producers and consumers critically interdepend on each other, particularly in the anaerobic digestion process [16,17].

The above studies have mostly focused on only the stability of microbial communities, but not their productivity. One exception is a modelling study on anaerobic digestion which recognized that a bistable state of the system could also be where an increased yield occurs [17]. In general, the productivity of a microbial community represents an important aspect of its function (e.g. [18]), which however is yet to be more generally understood for engineering applications together with stability. Besides, the systems analysed in the past commonly involve a single type of relationship between any two species in a community, e.g. competition or syntrophy, whereas in an artificially designed engineering system multiple relationships may coexist [19], which could be further complicated by the presence of interactions between chemical and biological species, for example, through feed or product inhibition. Therefore, an understanding of the impact of such more complex inter-connectedness is needed.

In this work, an in silico, generalized analysis is carried out on bioreactors based on synthetic microbial communities. The aim is to gain insights into how the structures and parametrization of such systems could affect their productivity and stability, to provide guidance to the engineering of such bioreactors. In particular, a base system with two cross-feeding species is used, on top of which a variety of two-species systems with inhibition(s) are constructed and studied. These systems, demonstrating the principle of division of labour by realizing a complex conversion task through a sequential process, represent the most common schemes of synthetic communities based bioreactors that have been reported in the literature (table 1). Furthermore, two representative three-species systems are investigated. These two- and three-species systems are described by generic mathematical models, and their behaviour and performance studied by bifurcation analysis and numerical simulation.

Table 1.

Literature examples of engineered cross-feeding microbial consortia.

ref.	feed	product	1st species	2nd species	inhibition
[20]	cellulose	hydrogen	C. cellulovorans, converting the feed to acids	R. palustris, converting acids to hydrogen	low pH, on both species, reduced by acids consumption of the 2nd species
[21]	cellulose	isobutanol	T. reesei, hydrolysing the feed to soluble carbohydrates	E. coli, converting soluble carbohydrates to product	isobutanol, on both species
[22]	cellulose	ethanol	C. phytofermentans, hydrolysing the feed to soluble carbohydrates	yeast, converting soluble carbohydrates to product	oxygen and cellodextrin (produced by 1st species), on the 1st species, reduced by consumption of both by the 2nd species
[5]	xylose	oxygenated taxanes	E. coli, converting xylose and producing taxadience and acetate	S. cerevisiae, consuming acetate and converting taxadience to product	acetate, on E. coli, reduced through consumption by the 2nd species

2. Methods

2.1. Construction of representative systems

To construct representative microbial communities, we consider a set of building blocks involving two parts: microbial species (denoted as X) and chemicals processed or produced by the species (denoted as S). With this notation, we first consider a base system that forms a simple cross (forward) feeding chain (figure 1), where species X₁ is produced by consuming chemical S₁ which itself is produced by species X₂ through consuming chemical S₂ that acts as the substrate of the overall system. In reality, X₁ would be the species to produce the final, desired chemical product. For the sake of simplicity, we assume such a product would have no feedback effect on the behaviour of the microbial community, therefore it is not explicitly represented in the model. Instead, the abundance of X₁ is assumed to be proportional to the quantity of the product and is taken as a proxy of the latter.

Figure 1.

The base system. (Online version in colour.)

On top of this base system, we construct its variations with three layers of complexity. In layer 1 (figure 2), four two-species cases (numbered as cases 1-1 to 1-4) are considered, each of which contains an inhibition between a chemical and a species or between the two species. In reality, an inter-species inhibition may still be mediated by a certain chemical, which is however not explicitly represented in the system structure. Note that, an inhibition between S₁ and X₁ and that between S₂ and X₂ are not considered, as they will not affect the interactions between the two species and are therefore not relevant to this study.

Figure 2.

Two-species systems with a single inhibition (layer-1 cases). (Online version in colour.)

In the next step, single inhibitions from layer 1 are combined, leading to six cases (numbered as cases 2-1 to 2-6), each of which includes two different inhibitions; these cases are referred as those of layer 2 (figure 3 and table 2). The layer-2 cases are considered in order to investigate the compound effects of multiple inhibitions that may exist in a two-species system.

Figure 3.

Two-species systems with double inhibitions (layer-2 cases). (Online version in colour.)

Table 2.

Relations between layer 1 and layer 2 models.

layer-2 case	layer-1 cases being combined
2-1	1-1, 1-2
2-2	1-1, 1-3
2-3	1-1, 1-4
2-4	1-2, 1-3
2-5	1-2, 1-4
2-6	1-3, 1-4

Finally, we introduce layer 3 where systems involving three species are considered. Compared to two-species systems, synthetic communities containing three (or more) species are still rare, albeit intuitively they have the potential to bring more opportunities for engineering design. On the other hand, the number of their possible structures is much larger due to the combinatorial effect, which would be difficult to investigate exhaustively. In this work, we take a view of incremental design and focus on the circumstances where a third species is introduced to a two-species system for a specific improvement. In particular, two important ecological functions of the third species are considered, namely (i) complementary resource utilization and (ii) facilitation. These considerations lead to the construction of two representative three-species cases for layer 3 (figure 4).

Figure 4.

Construction of the two cases of three-species system (layer-3 cases). (Online version in colour.)

The first one (case 3-1) is derived from the base system, now with X₂ producing not only S₁ but also S₃, as frequently encountered in industrial applications such as mixed-acid fermentation. To allow the additional intermediate product S₃ to be used, complementary to the utilization of S₁ by X₁, we further introduce the third species, X₃, to convert S₃ to a useful product (represented by X₃ itself). One of the complexities of engineering microbial communities that needs to be recognized when introducing a new species is the multiple effects of the species, which means it may cause unwanted complications (such as excreting a metabolite which is toxic to an existing species) while providing the intended function to the system. Reflecting this reality, we consider in case 3-1 that X₃ has a negative effect on X₂, modelled by inhibition.

The second three-species case (case 3-2) is constructed from case 1-3, with a newly introduced species X₃ to offer a facilitating function: easing the inhibition of X₂ on X₁. To incorporate this effect, we assume the inhibition is via a chemical S₃ produced by X₂ which negatively affects X₁, and the role of the new species, X₃, is to consume S₃, hence reducing the inhibition on X₂. Again, to include an unwanted side-effect for this case, we assume that the new species X₃ competes with X₁ for S₂.

2.2. Mathematical models

Well-mixed continuous bioreactors (i.e. chemostats) for the culture of multiple species are modelled with the commonly used Monod-type kinetics. For a two-species system, its mass balance is modelled by:

$$\frac{\mathrm{d}{X}_i}{\mathrm{d}t}=-D{X}_i+r_i,$$ 2.1

$$\frac{\mathrm{d}{S}_1}{\mathrm{d}t}=D(S_{1,\mathrm{in}}-S_1)+Y_{1,2}{r}_2-Y_{1,1}{r}_1,$$ 2.2

$$\frac{\mathrm{d}{S}_2}{\mathrm{d}t}=D(S_{2,\mathrm{in}}-S_2)-Y_{2,2}{r}_2$$ 2.3

$$\mathrm{and}D=\frac{1}{\mathrm{HRT}},$$ 2.4

where X_i represents the concentration of species i, i = 1,2; S_k is the concentration of chemical k, k = 1,2; S_k,in is the inlet concentration of chemical k; HRT and D are the hydraulic retention time and the dilution rate, respectively; Y_k,i is a yield factor, referring to the amount of chemical k that is produced or consumed by the growth of a unit amount of the biomass of microbial species i. The growth rates of the two species, r_i, are further modelled by

$$r_i={\mu }_{\max ,i}\frac{S_i}{S_i+K_i}{X}_i\prod I_{i,j},$$ 2.5

where, for species i, μ_max,i is the maximum specific growth rate, K_i is the half saturation constant. I_i,j represents the inhibition factor on species i by chemical or species j, and is modelled as follows:

$$I_{i,j}=\{\begin{array}{cc}\frac{1}{1+{\varphi }_{i,j}\times (X_j\hspace{0.17em}\hspace{0.17em}\mathrm{or}\hspace{0.17em}{S}_j)},& \mathrm{if}\hspace{0.17em}\mathrm{inhibition}\hspace{0.17em}\mathrm{is}\hspace{0.17em}\mathrm{present}\\ 1,& \mathrm{otherwise}\end{array}$$ 2.6

where i represents biological species being inhibited, ϕ_i,j is the inhibition constant.

To reduce the number of parameters, the above model is further converted to a dimensionless form. Below are equations for the dimensionless quantities to be discussed in the main text; other equations of the dimensionless model are given in appendix A.

$${\theta }_1={\mu }_{\max ,1}\times \mathrm{HRT},$$ 2.7

$${\theta }_2={\mu }_{\max ,2}\times \mathrm{HRT},$$ 2.8

$$\overline{X_1}=\frac{Y_{1,1}{X}_1}{K_1},$$ 2.9

$$\overline{X_2}=\frac{Y_{1,1}{X}_2}{K_1},$$ 2.10

$$\gamma =\frac{K_2}{K_1},$$ 2.11

$$\overline{t}=\frac{t}{\mathrm{HRT}},$$ 2.12

$${\alpha }_1=\frac{Y_{1,2}}{Y_{1,1}},$$ 2.13

$${\alpha }_2=\frac{Y_{2,2}}{Y_{1,1}}$$ 2.14

$$\mathrm{and}\overline{S_{2,\mathrm{in}}}=\frac{S_{2,\mathrm{in}}}{K_1}.$$ 2.15

The two three-species systems are modelled by extending the two-species model; details are given in appendix B.

2.3. Quantification of stability and productivity

Three aspects of a multi-species bioreactor, collectively referred to as its ‘stability behaviour’, are examined: (i) how fast the system which has deviated from a stable steady-state (with the coexistence of all species) following a disturbance can recover itself, that is, the rate of (re-)convergence to the previous steady-state; (ii) the size and position of the (productive) operating window, depicted by the range(s) of operating parameter(s), in which the system can reach a stable steady-state (i.e. with the coexistence of all species); and (iii) the occurrence of multiple steady states (i.e. bi-stability) or stable oscillation (i.e. stable periodic orbits).

To quantify the first aspect, the largest eigenvalue of the Jacobian matrix of a linearized dynamical system at a stable steady state is calculated, and its extent of negativity (i.e. how far it is below zero) is used as an indicator, referred to as ‘stability’ in the rest of the paper:

$$\mathrm{stability}=\hspace{0.17em}\max (|{\mathrm{Re}}_i|,i=1,2,\dots ,n),$$ 2.16

where Re_i is the real part of the ith of n eigenvalues; stability is calculated only for cases where max(Re_i, i = 1, 2, …, n) < 0.

The operating space of the bioreactor is represented by the range of the inlet concentration of substrate S₂, assuming S₁ is only produced internally by X₁; thus S₂ becomes the only feed to the bioreactor. For each of the cases in the three layers presented in §2.1, steady-state solutions are obtained analytically for each of the selected points in the operating space, the model is linearized at each steady-state solution, and eigenvalues of the Jacobian matrix of the linearized model are calculated. This thus forms the basis for calculating the stability indicator (equation (2.16)), as well as for quantifying the other two aspects of the stability behaviour. The operating window with stable coexistence corresponds to the range of inlet substrate concentration in which (i) each species has a positive steady-state concentration and (ii) the largest eigenvalue is negative. The multiplicity of stable steady states (i.e. bi-stability) can readily be detected by examining the steady-state solutions and the corresponding eigenvalues. The possible occurrence of a (stable) oscillatory behaviour is firstly identified by the detection of the eigenvalue with the largest real part becoming purely imaginary. Subsequently, dynamic simulation is carried out with parameter settings leading to such cases to confirm the occurrence of stable oscillation.

The productivity of a bioreactor at a steady state is defined as the amount of the produced output per unit amount of substrate supplied, and is quantified by:

$$\mathrm{productivity}=\hspace{0.17em}\frac{\overline{X_1}}{\overline{S_{2,\mathrm{in}}}}\hspace{0.17em}\mathrm{or}\hspace{0.17em}\frac{\overline{X_1}+\overline{X_3}}{\overline{S_{2,\mathrm{in}}}}$$ 2.17

depending on whether the bioreactor has X₁ as the only product or has both X₁ and X₃ as its products.

3. Results

3.1. The base system: two species with no inhibition

We start by analysing the behaviour and performance of the base system, where two species form a simple cross-feeding chain, without any inhibition effects. The model presented in §2.2 contains several kinetic- and yield-related parameters. Here, the variation in three parameters important for predicting the collective behaviour of the two species was considered: θ₁ and θ₂, representing the maximum amount of growth that can be achieved by the two species over a single hydraulic retention period; and α₁, as the ratio of yield factors of the two species that link the growth of the two species with the consumption or production of the intermediate substrate, S₁. As shown in table 3, eight sets of parameter values were tested, and the change in productivity and stability (as defined in §2.3) along with the inlet substrate concentration is predicted for each parameter set. For clarity, only part of the results is shown in figure 5 to demonstrate the main trends.

Table 3.

Sets of parameters tested for the base system.

parameter set	a	b	c	d	e	f	g	h
θ1	3	3	2	2	3	3	2	2
θ2	3	2	3	2	3	2	3	2
α1	1	1	1	1	1.5	1.5	1.5	1.5
γ1	1
α2	1

Figure 5.

Productivity (a) and stability (b) of the base system under different parameter settings. (Online version in colour.)

From figure 5a and b, one can see (simply by visual inspection) that for both productivity and stability, high values for θ₁, θ₂ and α₁ generally lead to improvement, with parameter sets ‘e’ (with highest values for all the three parameters) and ‘d’ (with lowest values for all the three parameters) corresponding to the best and the worst performance, respectively. Another common trend is that, for each parameter set, both productivity and stability of this base system appear to improve as the inlet substrate concentration increases. On the other hand, the difference in the parameter values is shown to cause variations in the lower bound of the inlet substrate concentration required for the coexistence of the two species (corresponding to the leftmost point on each curve in figure 5a and b; at an inlet substrate concentration lower than this point, washout occurs to one or both species). Again, higher parameter values appear to allow the coexistence region to start at a lower inlet substrate concentration, and because of this, parameter sets ‘e’ and ‘d’ represent the two extremes also in this respect. No multiple steady states or stable oscillations were detected for the base system.

3.2. Layer-1 cases: two species with one inhibition

The four cases in layer 1 are studied here. The results with parameter set ‘a’ are shown in figure 6; other parameter sets produced similar findings and are given in appendix C. Each case was evaluated with three levels of inhibition strength. Note, case 1-3 adopted lower values of the inhibition constant ϕ than those for the other cases, because higher inhibitions would in this case lead to the extinction of X₁ under any inlet substrate concentrations. Besides, as the inlet concentration varies, different eigenvalues of the system may become the one that has the largest real part. When this happens, the stability curve becomes unsmooth.

Figure 6.

Results for the layer-1 cases (θ₁ = 3, θ₂ = 3, α₁ = 1). (Online version in colour.)

From figure 6, it can be seen that, in all cases, an increase in the strength of a particular inhibition always leads to reduction in both productivity and stability. Case 1-1 (with the intermediate substrate inhibiting its producer) and case 1-4 (with the intermediate substrate inhibiting its consumer) demonstrate behaviours similar to the base system, in that the productivity and the stability generally both increase with the inlet substrate concentration.

In case 1-2 (with the downstream species X₁ inhibiting the upstream species X₂), when the inlet substrate concentration increases, X₂ tends to produce more S₁, and hence more X₁; however, the accumulation of X₁ will inhibit the activity of X_2, hence forming an indirect feedback loop. Beyond a certain inlet substrate concentration, the production of X₂ will begin to be reduced and may eventually stabilize at a moderate (but not diminishing) level. Furthermore, with this specific community structure and at high inhibition strengths, the increase in productivity is not always accompanied by the increase in stability, see for example the curves for $\overline{\varphi }=2$ in figure 6 when the inlet substrate concentration is larger than 2. When the inhibition effect is significant enough, the microbial community loses the ability to stabilize at a steady-state and starts to oscillate at high inlet substrate concentrations, as shown in figure 7, which is also a consequence of the feedback loop mentioned above.

Figure 7.

Oscillation in case 1-2. $\overline{\varphi }=5$, θ₁ = 3, θ₂ = 3 and α₁ = 1. (a) Productivity, solid line represents stable steady state, dashed line represents oscillation region; asterisk represents onset of stable oscillation. (b) Stability at stable steady state. (c) Numerical simulation of the stable oscillation, $\overline{S_{2,\mathrm{in}}}=\hspace{0.17em}6$. (Online version in colour.)

In case 1-3 (with the upstream species X₁ inhibiting the downstream species X₂), the most significant difference compared with the other cases is that, the inhibition effect of the upstream species on the downstream species in this case tends to eliminate the latter from the system, which occurs when the inlet substrate concentration is high. This means that an operating window with a limited size (as opposed to being open-ended) exists for case 1-3. Beyond the best inlet substrate concentration, both the productivity and the stability decline, until the downstream species ceases to survive. Besides, although the higher values of the three model parameters (θ_1, θ_2, α₁) lead to the improvement in productivity and stability across all the layer-1 cases (see appendix C), case 1-3 appears to be specifically sensitive to the change in parameter θ₁: when θ₁ becomes small, the uptake rate of S₁ decreases, and the inhibition affects the system to a greater extent. This makes the coexistence of the two species impossible at the highest inhibition strength for parameter sets ‘c’, ‘d’, ‘g’ and ‘h’ and also impossible at the medium inhibition strength for parameter sets ‘c’ and ‘d’, regardless of the level of the inlet substrate concentration.

As a final note on layer 1, the inhibition in cases 1-1 and 1-4 shows an additional effect (figure 6): the minimum inlet substrate concentration needed for coexistence increases in comparison with the base system, and the shift becomes greater with the increase in the inhibition strength.

3.3. Layer-2 cases: two-species with double inhibitions

As described earlier, the six layer-2 cases each contain two of the four individual inhibitions. Therefore, a primary concern of this part of the study is on the effect of the combination of two inhibitions. Since the study of the layer-1 cases has shown that applying different values for parameters θ₁, θ₂ and α₁ does not affect the qualitative characteristics of the sing-inhibition systems (appendix C), here only parameter set ‘a’ was applied together with inhibition parameters (table 4) to examine the layer-2 cases. The results are shown in figure 8.

Table 4.

Parameter values adopted in the layer-2 cases.

	case 2-1	case 2-2	case 2-3	case 2-4	case 2-5	case 2-6
inhibition from case 1-1 $({\overline{\varphi }}_1)$	2	1.5	0.8
inhibition from case 1-2 $({\overline{\varphi }}_2)$	0.6			0.2	1
inhibition from case 1-3 $({\overline{\varphi }}_3)$		0.3		0.3		0.2
inhibition from case 1-4 $({\overline{\varphi }}_4)$			0.2		1	0.5
θ1	3
θ2	3
α1	1
α2	1
γ1	1

Figure 8.

Results of layer-2 cases in comparison with corresponding layer-1 cases. (Online version in colour.)

The behaviour of the six layer-2 cases falls largely into two categories. In the first category, the two inhibitions impose their inhibitive effect on the system rather independently. It includes cases 2-1 (both S₁ and X₁ inhibiting X₂), 2-3 (S₁ inhibiting X₂ and S₂ inhibiting X₁), 2-5 (S₂ inhibiting X₁ and X₁ inhibiting X₂) and 2-6 (both S₂ and X₂ inhibiting X₁). In these cases, the combined effect of two inhibitions results in greater reduction in productivity and stability compared to the cases with any of the individual inhibitions, and may reduce the size of the operating window for coexistence as well (as in cases 2-3 and 2-6).

The two remaining layer-2 cases belong to the second category, where the two inhibitions are inter-connected, with the presence of one inhibition bringing an easing effect on the other. In case 2-4 (X₁ and X₂ inhibiting each other), this is achieved through a direct counter-action: the inhibition of X₁ to X₂ can give X₁ a better chance to survive the inhibition by X₂ at high inlet substrate concentrations. In case 2-2 (X₂ inhibiting X₁ and S₁ inhibiting X₂), the easing effect is realized through a negative feedback loop: reduced consumption of S₁ by X₁ following the inhibited growth of X₁ due to X₂ (first inhibition) can lead to the increased inhibition of X₂ by S₁ (second inhibition), due to the increased accumulation of S₁ resulting from the first inhibition. The presence of the second inhibition will reduce the growth of X₂, hence easing the first inhibition. This negative feedback loop thus allows X₁ to survive in this system at a high inlet substrate concentration which is impossible when only the first inhibition exists (i.e. as in case 1-3).

In addition to the aforementioned changes in productivity and stability, some of the layer-2 cases also demonstrate complex behaviours such as oscillation and bi-stability. The first ones to mention are cases 2-1, 2-4 and 2-5, all of which inherit the indirect feedback loop from case 1-2 which has been shown earlier to cause oscillation. All these three layer-2 cases were found to be able to give rise to oscillation at specific parameter settings, as shown in figure 9 for case 2-5 as an example. In case 2-3, on the other hand, two stable steady states emerge when the inlet substrate concentration is high (see figure 8, third row, left), which means a disturbance could cause the system to switch from the desirable steady state of coexistence and high productivity to the other one where X₁ is not able to survive. This bi-stability behaviour can be attributed to the positive feedback loop of X₂->S₂->X₁->S₁->X₂, a new structure formed due to the combination of two inhibitions, which can make the system drift away from a stable coexistence regime after it experiences a disturbance that causes a dip in X₂. The presence of bi-stability represents another situation where balanced consideration is needed on productivity and stability, as the bi-stability region is also where the productivity is higher.

Figure 9.

Oscillation in case 2-5. ${\overline{\varphi }}_2=2$, $\hspace{0.17em}{\overline{\varphi }}_4=0.2$. (a) Productivity, solid line represents stable steady state, dashed line represents oscillation region, asterisk represents onset of stable oscillation. (b) Stability at stable steady state. (c) Numerical simulation of the stable oscillation, $\overline{S_{2,\mathrm{in}}}=8.$ (Online version in colour.)

3.4. Layer-3 cases: addition of a third species

Here we study the two cases where a third species is introduced to the bioreactor for improving its performance, which however also brings some unwanted side effect to the microbial community. Same as for layer 2, parameter set ‘a’ has been used in the analysis. The complete set of parameter values for each of the layer-3 cases, together with their mathematical models, is given in appendix B.

As described earlier, case 3-1 augments the base system and involves a third species (X₃), which is introduced to use a second (and otherwise unused) intermediate product (S₃), which at the same time undesirably exhibits inhibition to the existing X₂. From figure 10, one can see that the change in the strength in the new inhibition alters the cost–benefit ratio. When the inhibition cost is sufficiently low, the new system can be more productive than the two-species counterpart, as expected. However, the introduction of the third species appears to reduce stability, regardless of the inhibition strength, and the change in stability with the inlet substrate concentration does not always synchronize with that in productivity. Structurally, case 3-1 resembles the combination of the base system and case 1-2; the latter was shown earlier (in §3.2) to exhibit oscillation when the inhibition from X₁ on X₂ is sufficiently high. In the three species system, this behaviour is again preserved, as shown in figure 11, which is similar to the layer-2 cases that contain the same structure.

Figure 10.

Three-species system, case 3-1. Solid line represents three species systems with different parameter settings, dashed line represents the base system. The dotted line corresponds to a dimensionless strength of 0.2 of the new inhibition; the solid line corresponds to a strength of 1. (Online version in colour.)

Figure 11.

Occurrence of stable oscillation in three-species system, case 3-1. The dimensionless strength of the new inhibition is 6. (a) Productivity, solid line represents stable steady state, dashed line represents oscillation region, asterisk represents onset of stable oscillation. (b) Stability at stable steady state. (c) Numerical simulation of the stable oscillation, $\overline{S_{2,\mathrm{in}}}=6$. (Online version in colour.)

In case 3-2, a third species (X₃) is introduced to case 2-3 to ease the inhibition from X₂ on X₁ by consuming the inhibiting metabolite S₃, but at the cost of X₃ competing with X₂ for S₁ (see appendix B for parameter settings). As shown in figure 12, with a high uptake rate by X₃ on S₃, this new species cannot co-survive with X₁ when the inlet substrate concentration is very low (below 1.5, see figure 12b). However, coexistence of all the three species becomes possible when the inlet substrate concentration increases, but the competition between X₃ and X₁ for metabolite S₁ results in a drop of productivity compared to the two-species system. The benefit of introducing the new species manifests only after the inlet substrate concentration becomes sufficiently high (over 4.5), where the gain by the removal of the inhibiting metabolite S₃ begins to overweigh the loss due to the competing consumption of the intermediate feed S₁.

Figure 12.

Three-species system, case 3-2. (a) Productivity; (b) stability. (Online version in colour.)

However, if the uptake rate on S₃ by X₃ is low, under no circumstances does the introduction of the third species demonstrate any advantages in productivity, as the cost always overweighs the benefit.

Finally, regardless of the level of the update rate on S₃, this second case of three species constantly shows reduced stability compared to its two-species counterpart, as observed from the first case in layer 3.

4. Discussion

The dynamics and stability of complex ecosystems have attracted considerable theoretical interests [23–26], with seemingly conflicting results presented on important issues such as the relationship between diversity and stability [27]. Different from these past studies which have commonly focused on natural systems involving potentially a large number of species, this work focuses on engineered microbial communities with rather limited sizes which are realistic for controlled construction. In particular, variations of the commonly encountered pattern of a two-species forward-feeding chain have been investigated by introducing one or two inhibitions and further by considering the introduction of a third species for complementarity (in resource utilization) and facilitation. In terms of stability that indicates how fast a system re-stabilizes to its previous steady state following a disturbance, all the two-species and three-species variations have shown a reduced stability compared to the simple forward-feeding chain. Some of the inter-species relations studied in this work can be regarded as competitive in the sense that the presence of one species imposes a negative impact on the growth of another species, e.g. the inhibition between X₁ and X₂. In [25], it was shown that the increase in competitive relationships in a (large) microbial community tends to increase the system's stability. Such a trend has not been observed in the systems studied in this work which are of much smaller sizes. For these systems, which are very much likely to be encountered in the design of bioreactors, the results of this work suggest a tendency of reduced stability as the complexity increases, which deserves attention in engineering decisions. In addition, a very recent study [28] proved the stability of several structures of microbial communities which are not covered by the present work. Nevertheless, it suggested that the stability of a community could be affected not only by its structure but also by the strength of the inter-species relationships, which is consistent with the finding of the current study.

In all the two-species cases studied in this work, inhibitions are shown to also reduce productivity. In the two three-species cases, intended gains in productivity have been observed only when the cost of introducing the third species is relatively moderate. Besides, the operating condition (represented by the inlet substrate concentration) that yields a higher productivity does not necessarily lead to a higher stability: the trends of productivity and stability responding to the increase in inlet substrate concentration may be different (such as in cases 1-2 and 3-1), and the highest productivity may correspond to an operating condition where bi-stability occurs (as in case 2-3). These all suggest a possible conflict between the two engineering goals.

As well as reducing the stability in the coexistence region, three out of the four types of individual inhibition have shown a negative impact on the size of the productive operating window. Besides, the involvement of one or two inhibitions, when forming an indirect negative or positive feedback loop, has shown the possibility of giving rise to oscillation or bi-stability, respectively, similar to the well-known behaviours that occur at the intracellular level due to biomolecular interactions [29,30]. When a parameter setting favourable for productivity falls in the operating window where stable oscillation is sustained, the bioreactor will be expected to operate in the naturally occurring cycles as opposed to a stable state. However, the presence of such inter-species structures does not necessarily lead to a sophisticated behaviour. For example, we mentioned earlier that an indirect negative feedback loop exists in case 2-2, yet no oscillation was identified in this case. This is possibly because the inhibition strength from X₂ on X₁ needs to be rather low in order to make coexistence feasible, and low strength inhibition does not render a sufficiently strong negative feedback needed for the oscillation to occur. This suggests that the occurrence of a complex behaviour in a synthetic microbial community (such as stable oscillation) may be determined by both its structural feature and the feasible parameter space.

There are existing studies on understanding the possibility of predicting the behaviour of a complex microbial community based on that of its basic elements. Using binary interactions to predict the dynamics of a system involving many species has previously been considered highly unreliable [19]. On the other hand, a recent study showed a significant level of accuracy in applying a simple rule to predict the coexistence of a microbial community with a large number of species based on the binary patterns [31]. In the present work, we have shown that the structural relationship between two separate inhibitions (being either ‘additive’ or ‘opposing’, cf. §3.3) can well predict the effect of combining the two inhibitions on productivity and stability. Besides, the inheritance of a special structural feature (e.g. indirect negative feedback) by a two-inhibition case from a one-inhibition case well predicts the preservation of a corresponding behaviour (e.g. oscillation). On the other hand, if a new (and important) structural feature occurs from the combination (e.g. a positive feedback, as in case 2-3), a new behaviour could emerge which cannot be predicted from the behaviour of the simpler systems. This suggests that for the synthetic microbial communities such as those studied in this work, the behaviour of a more complex system could be predicted, at least qualitatively, from that of its constituents, when there is no important new structure being formed as the complexity increases. However, even in such cases, detailed modelling analysis of the more complex system may still be needed for the purpose of engineering design, because quantitative (as opposed to qualitative) prediction could be important in order to shed light on issues such as productivity–stability trade-off, determination of the productive operational window, and cost–benefit analysis for introducing additional species.

Finally, since this work is confined to the analysis of bioreactors with a synthetic community that involves a small number of defined species, the dynamic evaluation of the community structure, often an interesting aspect of studying a community with a relatively large number of species, such as that in an anaerobic digester [32], has not been an important consideration here. Furthermore, the bioreactors have been assumed to be well-mixed; stability features arising from spatial heterogeneity [33] remain a subject of future work.

5. Conclusion

In this work, a number of representative structures of synthetic microbial community with two or three species have been evaluated in the context of a continuous bioreactor. Through bifurcation analysis and numerical simulation, it was shown that the introduction of inhibitions to a simple two-species cross-feeding chain generally reduces both productivity and stability of the bioreactor. When introducing a third species for complementary resource utilization or for facilitating existing species, productivity gain can occur when the benefit of the intended mechanism overweighs the cost of any negative side effect of the additional species, but accompanied with a reduction in stability. Inhibitions were also shown to be able to reduce the productive operating window that corresponds to the coexistence region of the operating space. Complex behaviours such as stable oscillation and bi-stability may occur with certain structural features such as feedback loops, combined with a suitable range of parameter values. It was also learned that the behaviour of a system that combines simpler structures may be qualitatively predicted based on the knowledge of its simpler constituents, particularly when no new important structural feature arises from the combination. However, detailed studies of the more complex systems, including both analytical stability analysis and numerical simulation, are warranted for engineering purposes in order to obtain the quantitative understanding needed to address issues such as the balance between productivity and stability of a bioreactor and between costs and benefits of introducing new species into an engineered microbial community.

Appendix A. Additional equations of the dimensionless model for the two-species system

$$\frac{\mathrm{d}\overline{X_1}}{\mathrm{d}\overline{t}}=-\overline{X_1}+\overline{r_1},$$

$$\frac{\mathrm{d}\overline{X_2}}{\mathrm{d}\overline{t}}=-\overline{X_2}+\overline{r_2},$$

$$\frac{\mathrm{d}\overline{S_1}}{\mathrm{d}\overline{t}}=(\overline{S_{1,\mathrm{in}}}-\overline{S_1})+{\alpha }_1\overline{r_2}-\overline{r_1},$$

$$\frac{\mathrm{d}\overline{S_2}}{\mathrm{d}\overline{t}}=(\overline{S_{2,\mathrm{in}}}-\overline{S_2})-{\alpha }_2\overline{r_2},$$

$$\overline{r_1}={\theta }_1\frac{\overline{S_1}}{\overline{S_1}+1}\overline{X_1}\overline{I_{i,j}},$$

$$\overline{r_2}={\theta }_2\frac{\overline{S_2}}{\overline{S_2}+\gamma }\overline{X_2}\overline{I_{i,j}},$$

$$\overline{s_1}=\frac{s_1}{K_1},$$

$$\overline{s_2}=\frac{s_2}{K_1},$$

$$\overline{t}=\frac{t}{\mathrm{HRT}},$$

$$\overline{S_{1,\mathrm{in}}}=\frac{S_{1,\mathrm{in}}}{K_1},$$

$$\overline{I_{i,j}}=\{\begin{array}{cc}\frac{1}{1+\overline{{\varphi }_{i,j}}\times (\overline{X_j}\hspace{0.17em}\mathrm{or}\hspace{0.17em}\overline{S_j})}& \mathrm{if}\hspace{0.17em}\hspace{0.17em}\mathrm{inhibition}\hspace{0.17em}\hspace{0.17em}\mathrm{is}\hspace{0.17em}\mathrm{present}\\ 1,& \mathrm{otherwise}\hspace{0.17em}\end{array}$$

and

$$\overline{{\varphi }_{i,j}}=\{\begin{array}{cc}\frac{{\varphi }_{i,j}{K}_1}{Y_{1,1}},& \text{for inhibitions introduced in cases 1-2 and 1-3}\\ {\varphi }_{i,j}{K}_1,& \mathrm{for}\hspace{0.17em}\hspace{0.17em}\mathrm{other}\hspace{0.17em}\mathrm{inhibitions}\hspace{0.17em}\end{array}\hspace{0.17em}$$

Appendix B. Mathematical models and parameter values for the three-species system

B.1. Model for case 3-1

$$\frac{\mathrm{d}{X}_1}{\mathrm{d}t}=\frac{1}{\mathrm{HRT}}(0-X_1)+{\mu }_{\max ,1}\frac{S_1}{S_1+K_1}{X}_1,$$

$$\frac{\mathrm{d}{X}_2}{\mathrm{d}t}=\frac{1}{\mathrm{HRT}}(0-X_2)+{\mu }_{\max ,2}\frac{S_2}{S_2+K_2}{X}_2{I}_{2,3},$$

$$\frac{\mathrm{d}{X}_3}{\mathrm{d}t}=\frac{1}{\mathrm{HRT}}(0-X_3)+{\mu }_{\max ,3}\frac{S_3}{S_3+K_3}{X}_3,$$

$$\frac{\mathrm{d}{S}_1}{\mathrm{d}t}=\frac{1}{\mathrm{HRT}}(S_{1,\mathrm{in}}-S_1)+Y_{1,2}{\mu }_{\max ,2}\frac{S_2}{S_2+K_2}{X}_2-Y_{1,1}{\mu }_{\max ,1}\frac{S_1}{S_1+K_1}{X}_1,$$

$$\frac{\mathrm{d}{S}_2}{\mathrm{d}t}=\frac{1}{\mathrm{HRT}}(S_{2,\mathrm{in}}-S_2)-Y_{2,2}{\mu }_{\max ,2}\frac{S_2}{S_2+K_2}{X}_2,$$

$$\frac{\mathrm{d}{S}_3}{\mathrm{d}t}=\frac{1}{\mathrm{HRT}}(S_{3,\mathrm{in}}-S_3)+Y_{3,2}{\mu }_{\max ,2}\frac{S_2}{S_2+K_2}{X}_2-Y_{3,3}{\mu }_{\max ,3}\frac{S_3}{S_3+K_3}{X}_3$$

$$\mathrm{and}{I}_{2,3}=\frac{1}{1+{\varphi }_{2,3}{X}_3}.\hspace{0.17em}$$

Note, I_2,3 represents inhibition to X₂ by X₃, and ϕ_2,3 is the corresponding inhibition constant.

Converting the above model to a dimensionless form:

$${\theta }_1={\mu }_{\max ,1}\times \mathrm{HRT},$$

$${\theta }_2={\mu }_{\max ,2}\times \mathrm{HRT},$$

$${\theta }_3={\mu }_{\max ,3}\times \mathrm{HRT},$$

$$\overline{X_1}=\frac{Y_{1,1}{X}_1}{K_1},$$

$$\overline{X_2}=\frac{Y_{1,1}{X}_2}{K_1},$$

$$\overline{X_3}=\frac{Y_{1,1}{X}_3}{K_1},$$

$${\gamma }_2=\frac{K_2}{K_1},$$

$${\gamma }_3=\frac{K_3}{K_1},$$

$$\overline{S_1}=\frac{S_1}{K_1},$$

$$\overline{S_2}=\frac{S_2}{K_1},$$

$$\overline{S_3}=\frac{S_3}{K_1},$$

$$\overline{t}=\frac{t}{\mathrm{HRT}},$$

$${\alpha }_{1,2}=\frac{Y_{1,2}}{Y_{1,1}},$$

$${\alpha }_{2,2}=\frac{Y_{2,2}}{Y_{1,1}},$$

$${\alpha }_{3,2}=\frac{Y_{3,2}}{Y_{1,1}},$$

$${\alpha }_{3,3}=\frac{Y_{3,3}}{Y_{1,1}},$$

$$\overline{S_{1,\mathrm{in}}}=\frac{S_{1,\mathrm{in}}}{K_1},$$

$$\overline{S_{2,\mathrm{in}}}=\frac{S_{2,\mathrm{in}}}{K_1},$$

$$\overline{I_{2,3}}=\frac{1}{1+\overline{{\varphi }_{2,3}}\overline{X_3}}$$

$$\mathrm{and}\overline{{\varphi }_{2,3}}=\frac{{\varphi }_{2,3}{K}_1}{Y_{1,1}}.$$

The differential equations now become:

$$\frac{\mathrm{d}\overline{X_1}}{\mathrm{d}\overline{t}}=-\overline{X_1}+{\theta }_1\frac{\overline{S_1}}{\overline{S_1}+1}\overline{X_1},$$

$$\frac{\mathrm{d}\overline{X_2}}{\mathrm{d}\overline{t}}=-\overline{X_2}+{\theta }_2\frac{\overline{S_2}}{\overline{S_2}+{\gamma }_2}\overline{X_2{I}_{2,3}},$$

$$\frac{\mathrm{d}\overline{X_3}}{\mathrm{d}\overline{t}}=-\overline{X_1}+{\theta }_3\frac{\overline{S_3}}{\overline{S_3}+{\gamma }_3}\overline{X_1},$$

$$\frac{\mathrm{d}\overline{S_1}}{\mathrm{d}\overline{t}}=(\overline{S_{1,\mathrm{in}}}-\overline{S_1})+{\alpha }_{1,2}{\theta }_2\frac{\overline{S_2}}{\overline{S_2}+{\gamma }_2}\overline{X_2}-{\theta }_1\frac{\overline{S_1}}{\overline{S_1}+1}\overline{X_1},$$

$$\frac{\mathrm{d}\overline{S_2}}{\mathrm{d}\overline{t}}=(\overline{S_{2,\mathrm{in}}}-\overline{S_2})-{\alpha }_{2,2}{\theta }_2\frac{\overline{S_2}}{\overline{S_2}+{\gamma }_2}\overline{X_2}$$

$$\mathrm{and}\frac{\mathrm{d}\overline{S_3}}{\mathrm{d}\overline{t}}=(\overline{S_{3,\mathrm{in}}}-\overline{S_3})+{\alpha }_{3,2}{\theta }_2\frac{\overline{S_2}}{\overline{S_2}+{\gamma }_2}\overline{X_2}-{\alpha }_{3,3}{\theta }_3\frac{\overline{S_3}}{\overline{S_3}+{\gamma }_3}\overline{X_3}.\hspace{0.17em}$$

B.2. Model for case 3-2

$$\frac{\mathrm{d}{X}_1}{\mathrm{d}t}=\frac{1}{\mathrm{HRT}}(0-X_1)+{\mu }_{\max ,1}\frac{S_1}{S_1+K_1}{X}_1{I}_{1,3},$$

$$\frac{\mathrm{d}{X}_2}{\mathrm{d}t}=\frac{1}{\mathrm{HRT}}(0-X_2)+{\mu }_{\max ,2}\frac{S_2}{S_2+K_2}{X}_2,$$

$$\frac{\mathrm{d}{X}_3}{\mathrm{d}t}=\frac{1}{\mathrm{HRT}}(0-X_3)+{\mu }_{\max ,3,3}\frac{S_3}{S_3+K_{3,3}}{X}_3+{\mu }_{\max ,3,1}\frac{S_1}{S_1+K_{3,1}}{X}_3,$$

$$\frac{\mathrm{d}{S}_1}{\mathrm{d}t}=\frac{1}{\mathrm{HRT}}(S_{1,\mathrm{in}}-S_1)+Y_{1,2}{\mu }_{\max ,2}\frac{S_2}{S_2+K_2}{X}_2-Y_{1,1}{\mu }_{\max ,1}\frac{S_1}{S_1+K_1}{X}_1-Y_{1,3}{\mu }_{\max ,3,1}\frac{S_1}{S_1+K_{3,1}}{X}_3,$$

$$\frac{\mathrm{d}{S}_2}{\mathrm{d}t}=\frac{1}{\mathrm{HRT}}(S_{2,\mathrm{in}}-S_2)-Y_{2,2}{\mu }_{\max ,2}\frac{S_2}{S_2+K_2}{X}_2,$$

$$\frac{\mathrm{d}{S}_3}{\mathrm{d}t}=\frac{1}{\mathrm{HRT}}(S_{3,\mathrm{in}}-S_3)+Y_{3,2}{\mu }_{\max ,2}\frac{S_2}{S_2+K_2}{X}_2-Y_{3,3}{\mu }_{\max ,3,3}\frac{S_3}{S_3+K_{3,3}}{X}_3$$

$$\mathrm{and}{I}_{1,3}=\frac{1}{1+{\varphi }_{1,3}{S}_3}.$$

μ_max,3,3 and μ_max,3,1 are maximum growth rate of X₃ by uptake of S₃ and S₁, respectively, and K_3,3 and K_3,1 are the corresponding half-saturation constants. In this case, we assume the growth of X₃ is increased by uptake of S₁ and S₃ at the same time, in other words, uptake of either substrates is independent and without priority.

Converting the above model to a dimensionless form:

$${\theta }_1={\mu }_{\max ,1}\times \mathrm{HRT},$$

$${\theta }_2={\mu }_{\max ,2}\times \mathrm{HRT},$$

$${\theta }_{3,1}={\mu }_{\max ,3,1}\times \mathrm{HRT},$$

$${\theta }_{3,3}={\mu }_{\max ,3,3}\times \mathrm{HRT},$$

$$\overline{X_1}=\frac{Y_{1,1}{X}_1}{K_1},$$

$$\overline{X_2}=\frac{Y_{1,1}{X}_2}{K_1},$$

$$\overline{X_3}=\frac{Y_{1,1}{X}_3}{K_1},$$

$${\gamma }_2=\frac{K_2}{K_1},$$

$${\gamma }_{3,1}=\frac{K_{3,1}}{K_1},$$

$${\gamma }_{3,3}=\frac{K_{3,3}}{K_1},$$

$$\overline{S_1}=\frac{S_1}{K_1},$$

$$\overline{S_2}=\frac{S_2}{K_1},$$

$$\overline{S_3}=\frac{S_3}{K_1},$$

$$\overline{t}=\frac{t}{\mathrm{HRT}},$$

$${\alpha }_{1,2}=\frac{Y_{1,2}}{Y_{1,1}},$$

$${\alpha }_{2,2}=\frac{Y_{2,2}}{Y_{1,1}},$$

$${\alpha }_{3,2}=\frac{Y_{3,2}}{Y_{1,1}},$$

$${\alpha }_{3,3}=\frac{Y_{3,3}}{Y_{1,1}},$$

$${\alpha }_{1,3}=\frac{Y_{1,3}}{Y_{1,1}},$$

$$\overline{S_{1,\mathrm{in}}}=\frac{S_{1,\mathrm{in}}}{K_1},$$

$$\overline{S_{2,\mathrm{in}}}=\frac{S_{2,\mathrm{in}}}{K_1},$$

$$\overline{I_{1,3}}=\frac{1}{1+\overline{{\varphi }_{1,3}}\overline{S_3}}$$

$$\mathrm{and}\overline{{\varphi }_{1,3}}={\varphi }_{1,3}{K}_1.$$

The differential equations now become:

$$\frac{\mathrm{d}\overline{X_1}}{\mathrm{d}\overline{t}}=-\overline{X_1}+{\theta }_1\frac{\overline{S_1}}{\overline{S_1}+1}\overline{X_1}\overline{I_{1,3}},$$

$$\frac{\mathrm{d}\overline{X_2}}{\mathrm{d}\overline{t}}=-\overline{X_2}+{\theta }_2\frac{\overline{S_2}}{\overline{S_2}+{\gamma }_2}\overline{X_2},$$

$$\frac{\mathrm{d}\overline{X_3}}{\mathrm{d}\overline{t}}=-\overline{X_1}+{\theta }_{3,3}\frac{\overline{S_3}}{\overline{S_3}+{\gamma }_{3,3}}\overline{X_1}+{\theta }_{3,1}\frac{\overline{S_1}}{\overline{S_1}+{\gamma }_{3,1}}\overline{X_3},$$

$$\begin{array}{rl}\frac{\mathrm{d}\overline{S_1}}{\mathrm{d}\overline{t}}=& (\overline{S_{1,\mathrm{in}}}-\overline{S_1})+{\alpha }_{1,2}{\theta }_2\frac{\overline{S_2}}{\overline{S_2}+{\gamma }_2}\overline{X_2}-{\theta }_1\frac{\overline{S_1}}{\overline{S_1}+1}\overline{X_1}\\ & -{\alpha }_{1,3}{\theta }_{3,1}\frac{\overline{S_1}}{\overline{S_1}+{\gamma }_{3,1}}\overline{X_3},\end{array}$$

$$\frac{\mathrm{d}\overline{S_2}}{\mathrm{d}\overline{t}}=(\overline{S_{2,\mathrm{in}}}-\overline{S_2})-{\alpha }_{2,2}{\theta }_2\frac{\overline{S_2}}{\overline{S_2}+{\gamma }_2}\overline{X_2}$$

$$\mathrm{and}\frac{\mathrm{d}\overline{S_3}}{\mathrm{d}\overline{t}}=(\overline{S_{3,\mathrm{in}}}-\overline{S_3})+{\alpha }_{3,2}{\theta }_2\frac{\overline{S_2}}{\overline{S_2}+{\gamma }_2}\overline{X_2}-{\alpha }_{3,3}{\theta }_3\frac{\overline{S_3}}{\overline{S_3}+{\gamma }_{3,3}}\overline{X_3}.\hspace{0.17em}$$

B.3. Parameter values adopted

Table 5.

Parameter values adopted by the two layer-3 cases.

case	parameters	value
case 3-1	γ2, γ3	1
	θ1	3
	θ2	3
	θ3	1
	α1,2	1
	α2,2	1
	α3,3	1
	α2,3	0.5
	${\overline{\varphi }}_{2,3}$	various, see text
case 3-2	γ2, γ3,1, γ3,3	1
	θ1	3
	θ2	3
	θ3,1	1
	θ3,3	1
	α1,2	1
	α2,2	1
	α3,2	1
	α1,3	1
	α3,3	1
	${\overline{\varphi }}_{1,3}$	0.3

Appendix C. Results of layer-1 cases with different sets of parameter values

In the figures 13–16, lines in different patterns represent different inhibition strength, same as in figure 6. In each figure, the sub-plots a-h correspond to different θ₁, θ₂, α₁ values, as given in table 2.

Figure 13.

Case 1-1. (Online version in colour.)

Figure 14.

Case 1-2. (Online version in colour.)

Figure 15.

Case 1-3. (Online version in colour.)

Figure 16.

Case 1-4. (Online version in colour.)

Data accessibility

This article has no additional data.

Authors' contributions

S.D. carried out the work, participated in the design of the study and drafted the manuscript; A.Y. conceived of the study, designed the study, and helped draft the manuscript. All authors gave final approval for publication.

Competing interests

We declare we have no competing interests.

Funding

We received no funding for this study.