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The present study combines the theory and the experimental data to predict the changes on intestinal bacterial populations during ingestion of beneficial probiotic bacteria. Our proposed model is a modified version of the Lotka-Volterra model, which takes the probiotic administration into account. Using the linear stability analysis of the model, the conditions for coexistence of the probiotics with other bacteria are established. Using the model fitted to the data of C. coccoides species and Bifidobacterium species, the effects of oral probiotics on autochthonous bacterial cultures is investigated. The estimated parameter values suggest that C. coccoides and Bifidobacterium facilitate each other during the probiotics administration, whereas they compete in the absence of the probiotics administration. This may suggest the beneficial effect of probiotic administration as it promotes the growth of C. coccoides species. The results also confirm prior studies showing that once probiotic supplementation is discontinued, the probiotic population and the promoting effect within the digestive tract will diminish.

Probiotics are live microorganisms which are thought to confer a health benefit on the host, when administered in adequate amounts [

Despite the above-mentioned benefits of probiotics, some studies suggest that probiotics may actually have damaging effects in certain cases. For instance, some infants who received Lactobacillus developed sepsis [

Given the benefits and harms of probiotics, there is a strong need to unpack the underlying mechanisms governing the interactions between probiotics and intestinal bacteria. Using a mathematical modeling approach, the main objective of the present work is to investigate the effects of probiotics administration on the microbial ecology of the intestine. To achieve this goal, we focus on a group of probiotics with the genus Bifidobacterium. Previous studies suggest that certain dosage of Bifidobacterium may positively influence human health [^{8} live Bifidobacterium cells helped alleviate many symptoms associated with Irritable Bowel Syndrome, the same team found that 10^{6} live cells and 10^{10} live cells actually exacerbated the same symptoms [^{10} probiotics (the mixture included but was not limited to Bifidobacterium) to the diet of these patients actually increased their mortality rate [

Patients with Irritable Bowel Syndrome and patients with Infectious Colitis exhibit very similar deviations from gut bacteria homeostasis when compared with healthy patients. Both Clostridium coccoides and Bifidobacterium populations are suppressed in the afflicted patients when compared with healthy subjects [

Using a mathematical modeling approach and the collected data, this paper investigates the potential interactions between the Bifidobacterium and C. coccoides species, and we posit that such interactions exist because several studies suggest that bacteria populations within the intestines interact with each other [

In the present work, C. coccoides species was selected because several studies have also used the Erec482, C. coccoides group, in human and animal studies [

The rest of this paper is organized as follows. Section 2 provides details of data collection, model construction, model fitting, and analysis of the model. Section 3 provides the main finding of the present work including the possible outcomes of the model and prediction of the interactions between the species both in the presence and absence of probiotics administration. Section 4 provides a discussion of the results and delivers the main conclusions of this study.

The present study combines the theory and the experimental data to predict the changes on intestinal bacterial populations during ingestion of beneficial probiotic bacteria. The temporal data of C. coccoides and Bifidobacterium species are collected before, during, and after probiotic (i.e., Bifidobacterium species) administration. Using a Lotka-Volterra Modeling approach, a mathematical model of probiotics and intestinal bacteria is constructed. The model is analyzed to determine the conditions for existence and stability of equilibria. The model is also fitted to data to determine the interaction between the species and to provide quantitative estimates of intestinal bacteria in response to probiotic administration.

A healthy Schnauzer adult dog received 2 tablets (2 times 10^{8} cfu (numbers of bacteria) of Bifidobacterium species) of Prostora® daily for a total of 4 days. During the 10 days of this study, the dog defecated approximately 30 grams of feces per day (~15 grams in the morning and ~15 grams at night). Fecal samples were collected before probiotic administration (Days 0, 1, and 2), during probiotic administration (Days 3, 4, 5, and 6) and after probiotic administration (Days 7, 8, and 9). Total fecal bacteria and two different fecal bacterial groups (i.e., the C. coccoides group and the probiotic group) were quantified in feces using fluorescent in situ hybridization. This technique relies on the bounding of fluorescently-labeled oligonucleotides probes to specific RNA sequences of the bacterial ribosomal RNA. This bounding allows the visualization and quantification of microorganisms by means of fluorescent detection.

Previous mathematical models for probiotic (in this case, Bifidobacterium and Lactobacillus) intervention have found it necessary to include parameters which

express the potential negative effects of probiotics upon the host organism by a degradation of the integrity of the intestinal wall [

Before and after During

probiotic administration probiotic administration

where the population growth of species i, carrying capacity of species i and interactions between the species i and j are denoted by

Using direct calculations and a geometric argument, the equilibrium solutions of model (1) were determined both in the presence and absence of probiotics administration. By linearizing model (1) about each equilibrium, the conditions for stability of each equilibrium were determined. The stability of the coexistence equilibrium was numerically verified for different sets of parameter values. Finally, using the Matlab optimization toolbox (the function fminsearch. m), mo- del (1) was fitted to the data and the specific parameter values were determined.

Since variables A(t), C(t), and P(t) are bacterial population, we have

Symbol | Description |
---|---|

A(t), C(t), P(t) | Model Variables; Population of Given Bacteria Group |

Growth Parameters; Growth Rate of Given Bacteria Group | |

Growth Rate Divided by the Carrying Capacity Interaction Parameters; (the subscript shows the interactive impact of first bacteria on second bacteria) | |

f, h, g α, d | Beneficial Effect of Paired Bacteria on Given Bacteria; Probiotic Ingestion Rate, and Probiotic Dissolving Rate |

Note: The parameters indicated in the last two rows are experimental parameters, which are set to zero before and after administration

where

Suppose that

where

In an unrealistic case, we may consider

tion

near stability analysis of these equilibria is given in Appendix A.

As shown in Appendix B, for

When

Theorem 1. Consider the system

Model Outcome | Equilibrium | Required Existence and Stability Conditions^{(1)} |
---|---|---|

Extinction | ||

Probiotics dominance | ||

C. coccoides dominance | ||

Coexistence | ||

Founder^{(2)} Control | Either | |

Founder^{(2)} Control | Either |

Notes: ^{(1)}the symbol ~ indicates that one of the following conditions must be violated; ^{(2)}depending on the initial conditions, the solution may converge to either equilibrium

Model Outcome | Equilibrium | Required Existence and Stability Conditions^{(1)} |
---|---|---|

Probiotics dominance | ||

Coexistence |

Note: There can be up to two coexistence equilibria,

1) the eigenvalues λ_{k} of A,

2)

then the solutions of the system are bounded and

Proof: See ( [

Theorem 2. Consider the system

1) A is a constant matrix with eigenvalues

2)

then for all solutions of the system, we have

where

side of model (2) with

In system (5), by substituting the linearization

where

The general solution of system (6) is of the form

Parameter | Before & After | During | Parameter | Before & After | During |
---|---|---|---|---|---|

5.0898 | 5.0898 | −0.0339 | −0.0339 | ||

−0.0723 | −0.0723 | 0 | 0.9059 | ||

2.3896 | 2.3896 | 0 | 0.8508 | ||

0.4116 | 0.4116 | 0 | 0.1627 | ||

0.3506 | 0.3506 | 0 | 0.8057 |

Notes: The Sum of the Squared Error (SSE) was 27.6153 for before and after probiotics administration and 88.6003 during the administration. The negative value of

After running MATLAB’s ODE45 and fminsearch. m, the parameter estimations yielding the lowest error were calculated for two cases of presence and absence of probiotics administration.

The main objective of this study was to compare the changes in the parameter values before, after, and during the experiment. The primary parameters of interest are

Additionally, this relationship appears to be amplified in the presence of Bifidobacterium supplementation. The parameter g which denotes C. coccoides’ beneficial effect upon Bifidobacterium is significantly greater than h which signifies Bifidobacterium’s beneficial effect upon C. coccoides. Thus, it seems that C. coccoides overall assists Bifidobacterium’s population growth while Bifidobacterium is essentially ambivalent about C. coccoides.

Further of note is that the solution curves of the model indicate that C. coccoides and Bifidobacterium populations move in tandem. Their highs and lows coordinate very well, so they seem to be responding to the same stimulus for growth and decay. However, this study is unable to go into causal factors for why this correlation relationship exists.

Also, despite the fact that our parameter estimations seem to indicate that C. coccoides and Bifidobacterium have beneficial effects upon each other, it should be noted that in the raw data, C. coccoides actually decreases throughout the observation period. This could be due to the residual effects of the supraphysiological levels of Bifidobacterium given during administration and the high

In conclusion, the present study suggests that Bifidobacterium and C. coccoides populations move nearly simultaneously and with similar magnitudes. Also, the parameter estimations imply that C. coccoides assist Bifidobacterium populations much more so than Bifidobacterium assist the C. coccoides population. However, further studies are likely needed in order to examine the after supplementation effects of Bifidobacterium administration and how the two population groups interact once supplementation has ceased.

Brown, T., Bani-Yaghoub, M. and García-Mazcorro, J.F. (2017) Understanding the Competitive and Cooperative Interactions between Probiotics and Autochthonous Intestinal Bacteria. Journal of Biosciences and Medicines, 5, 63-80. https://doi.org/10.4236/jbm.2017.54007

The Model is given by:

There are four equilibria:

The Jacobian matrix is given by:

Evaluating the Jacobian matrix at the first equilibrium,

which gives the eigenvalues

Hence,

Similarly, for the probiotics-free equilibrium, we have

So, we need to have:

Additionally, for the C. coccoides-free equilibrium, we have

which gives the eigenvalues

So, we need to have:

To determine the stability conditions for the coexistence equilibrium

which has the corresponding Jacobian matrix:

If

If

We have

Also, we require that

There are two cases:

If

and

But (2) and (3) imply that

If

and

which implies

In summary, (i)

Or

Moreover,

2)

The model is given by:

There are only two possible equilibria:

1)

2)

Substitute (13) into (12) and set equal to zero.

We get that

We need to have

There are four possibilities.

1)

2)

3)

Suppose that (14) has a real positive root

Stability of

If we compare model (11), (12) with model (1), (2) on page 1, we get that the Jacobian matrix of model (11), (12) is the same as that of model (1), (2) except for the following changes:

Hence, following the same procedure, we get that

We also get that

eq(12):

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