They are everywhere. Some 100 trillion inhabit the earth, comprising half of the animal mass on it. Have you guessed what I am talking about yet? See the following articles in the New York Times, NY Times Magazine, Scientific American, Nature, Science, or this TED talk to refresh your memory. Now the human microbiome has been associated with almost every disease possible, microbes in the gut have even been associated with brain diseases [1]. The study of these little things is kind of a big deal.

What is a normal human microbiome?
The most important developments in the human microbiome have come via the analysis of large cohorts across body sites (gut, mouth, vagina, skin, etc) [2] and longitudinal studies where fecal samples have been collected on a daily scale [3,4]. What we know from these studies is that the abundance and kinds of microbes are body site specific. See the figure just below that illustrates this point [2, Figure 1c].

In the figure above 4,788 specimens from 242 adults are projected into the first two principle coordinates (relative abundance of microbes at genus level). The different body sites are color coded, and it is clear that the specimens cluster according to body site and not by subject. We have also learned that microbial abundances are fairly stable for each site and for each subject (I will discuss this in more detail shortly). Before getting to the dynamics and estimation part we need a story so as to understand the translational implications of a better understanding of the human microbiome.

Fecal Microbial Transplantation
This story begins with Jane coming to the hospital because of an infection in her leg. To kill the infection she is given broad spectrum antibiotics. After a few days the infection is gone, but Jane now has severe diarrhea. The antibiotics have killed some of the healthy bacteria in her gut and now Jane has an over abundance of Clostridium difficile, i.e. she has Clostridium Difficile Infection (CDI). Ironically the most often prescribed treatment for CDI is another antibiotic. This targeted antibiotic always works in temporarily reducing the abundance C. difficile, but the CDI is recurrent. So with no other options Jane asks her brother John for a fecal sample. This fecal sample is prepared and transplanted into Jane (Fecal Microbial Transplantation (FMT)). As if a miracle has occurred Jane is healthy again. This kind of story is becoming commonplace in hospitals around the country now.

What happens in terms of the abundance of the microbes post-FMT is quite amazing. Consider the figure below [5, Figure 1]. What is being illustrated is several subjects stool samples pre-FMT, circled in red, and the trajectories (post-FMT) are shown to rapidly converge to the green circle (which also contains the host sample), overlaid on top of the samples from the 242 healthy adults above. A movie of these trajectories can be downloaded here. While the post-FMT samples do deviate slightly from the host sample in terms of relative abundance of microbes the stool samples remain within the range of what is considered a healthy stool sample. Stated simply what we are observing is the patient’s gut microbiome reconstituted and remaining in an abundance profile similar to that of the donor. It is quite amazing.

Is the microbiome stable?
What we just saw above was that the post FMT stool samples remained similar to the host after transplantation. So then one natural question arises: How stable is the human microbiome? Even biologist recognize that this is an important subject as is evidenced by the figure below which appeared in a recent review article in Science [6, Figure 1]

While I am delighted to see that the notion of stability has been recognized as an important issue in the human microbiome, I have felt a push back from the microbiome research community to explore what this actually implies. There is also a misunderstanding of what the word stable means. This is simply an ignorance issue and as control engineers/theorist we should just simply educate those in this field. Consider the figure below [7, Figure 3A] that shows 15 days of samples (shown in yellow) taken from the daily 1 year gut microbiome study in [3], and projected onto the principle coordinates from a previous study [8,9]. Ignore the red, green, and blue dots and focus on the trajectories of the yellow dots with gray lines following the day to day changes in the stool samples. The authors of [7] wanted to highlight the fact that the samples can deviate from steady state in almost all directions. The authors unfortunately draw the conclusion that this is a visualization of instability in the gut microbiome. The original figure from the study in [7] is shown on the left and my annotated figure is on the right. I would like to illustrate that the two trajectories after deviation return to the “steady region’’. This is not instability, but the very definition of stability. One could even argue we are observing asymptotic like stability, i.e. in the absence of disturbances all trajectories converge to a single fixed point. Of course there will always be disturbances in biological systems. Could this line of reasoning help to explain the success of FMT? I think you can begin to see where those working in the area of dynamics and control might be needed in this emerging field.

How do we model the microbiome?
The most common way that microbes interact is through the consumption of nutrients and the synthesis of products (not necessarily through the direct consumption of each other) [10]. Therefore, a detailed model would contain states for both the abundance of microbes and the abundance of the metabolites they consume and synthesize. At the finest level of modelling all host and microbe metabolic pathways would need to be mapped. We currently do not posses the technology or sufficiently rich data to perform this rigorously. At this point in our understanding of microbial dynamics it is more common to think of a reduced order model that only accounts for the abundances of the microbes.

The two most popular (reduced order) models are Generalized Lotka-Volterra (GLV) dynamics over a network and Bayesian networks. The first is deterministic (most common as well) and the second, probabilistic. I will focus on the first one here, but a similar discussion could follow with a probabilistic mind set as well, just a lot more capital E’s.

Let $$x_i$$ be the abundance of microbe $$i$$ for subject $$X$$ at a specific location on/in the body. Let’s assume for now we are concerned only with the gut. Then the GLV model for $$n$$ microbes interacting in the gut of subject $$X$$ is described by the following differential equation

$\dot{x}_i=r_ix_i+\sum_{j=1}^na_{ij}x_j$ where $$i=1,2,…,n$$. Collecting the abundances of the microbes into a column vector $$x=[x_1,\ x_2,\ \ldots ,\ x_n]^T$$ the dynamics can be compactly written as $$\dot{x}=\text{diag}(r)x+\text{diag}(x)Ax,$$
where $$r$$ is a column vector of the $$r_i$$ and $$[A]_{ij}:=a_{ij}$$. We will refer to $$A$$ as the microbial interaction matrix, or network. In this modeling paradigm $$r$$ captures the linear growth or death terms and the matrix $$A$$ captures the causal interactions amongst species. Thus the entry $$a_{ij}$$ is determined by thinking of the average affect that species $$j$$ has on species $$i$$ by determining what species $$j$$ generates as products and what both species $$i$$ and $$j$$ consume as nutrients. For instance, if species $$j$$ produces products that species $$i$$ consumes as nutrients and they do not compete for any other nutrients then the entry of $$a_{ij}$$ would be positive. I mentioned earlier that we dot not fully understand the microbial and metabolic interactions well enough to have a global bottom up model. Do we have sufficient data to learn the interactions in the simplified GLV model? We will discuss this in more detail shortly. Note in this blog the term “species’’ is in the general context of ecology, i.e. a set of organisms adapted to a particular set of resources in the environment, unless we state that we are specifically discussing the taxonomic rank “species’’.

Lets now consider the gut of a different individual, subject $$Y$$, and assume that the dynamics are as follows

$\dot y = \text{diag}(\bar r) y + diag(y) \bar A y.$
Notice that I have written the dynamics for both subjects with different variables, $$(r,A)$$ for subject $$X$$ and $$(\bar{r},\bar{A})$$ for subject $$Y$$. Is it possible that for two otherwise healthy individuals with similar diet $$A=\bar{A}$$, and $$r=\bar{r}$$. Recent attempts to infer the interaction matrices for two individuals illustrates some short comings in the literature and another opportunity for those working in system identification and machine learning to have an immediate impact in this field.

Consider the networks just below illustrating a subset of the interaction matrix for two subjects gut microbes [11, Figure 6]. This study concludes that the network of causal interactions between microbes are not the same for healthy individuals. There are many issues with this study however, illustrating the need for those working in the area of system identification to collaborate with those working on the human microbiome. I do not want to disparage the authors of [11], my only intention here is point out mistakes in the analysis that a control engineer might have noticed.

The authors correctly recognize that the data was not sufficiently rich (1 year of daily samples with not very much excitation) to accurately capture all species interactions (on the order of 100 at the taxonomic rank of species). Thus, the authors concluded to perform system identification using only the 14 most abundant species, and then showed that the two networks are different. One will realize however that the throwing away of states is problematic, and not the appropriate way to overcome a lack of sufficient richness. My own work is in fact pointing to the opposite scenario, otherwise healthy individuals have the same underlying interaction network, but I will withhold claiming that until I have more proof.

Our help is also needed in helping the physicians design their trials so that samples can be obtained with as much information as possible. There is also the technical issue of microbial samples usually being normalized (we only know relative abundances with confidence). System identification in biological networks is often referred to as network reconstruction, and this entire sub area of biological research is in very serious need of our help as this scathing comment in nature biotechnology points out.

Lots of open questions

• Are some body sites more stable than others?
• How do we rigorously demonstrate this stability?
• Are the networks of two healthy individuals similar?
• How do different diseases affect that network?
• Why do FMTs work?
• Are there other modelling approaches that can be used to understand microbial dynamics?
• What are the fundamental limitations for network reconstruction when dealing with relative abundances?
• Finally, how do we control the microbiome?

Conclusions
Aircraft control has been one of the cornerstone applications for control for more than 50 years. It is time however to find new areas for research. I hope this has inspired you to consider some translational areas such as the human microbiome as a possible research area for the application of everything you have learned in dynamics, control, and system identification.

I would like to acknowledge my collaborators Yang-Yu Liu and Amir Bashan, as well as conversations I have had with Aimee Milliken, Eric Alm, Curtis Huttenhower, and Rob Knight.

Biblipgraphy

1. Mayer, Emeran A., et al. “Gut microbes and the brain: paradigm shift in neuroscience.” The Journal of Neuroscience 34.46 (2014): 15490-15496.
2. Human Microbiome Project Consortium. “Structure, function and diversity of the healthy human microbiome.” Nature 486.7402 (2012): 207-214.
3. Caporaso, J. Gregory, et al. “Moving pictures of the human microbiome.”Genome Biol 12.5 (2011): R50.
4. David, Lawrence A., et al. “Host lifestyle affects human microbiota on daily timescales.” Genome Biol 15.7 (2014): R89.
5. Weingarden, Alexa, et al. “Dynamic changes in short-and long-term bacterial composition following fecal microbiota transplantation for recurrent Clostridium difficile infection.” Microbiome 3.1 (2015): 10.
6. Costello, Elizabeth K., et al. “The application of ecological theory toward an understanding of the human microbiome.” Science 336.6086 (2012): 1255-1262.
7. Knights, Dan, et al. “Rethinking “enterotypes”.” Cell host & microbe 16.4 (2014): 433-437.
8. Arumugam, Manimozhiyan, et al. “Enterotypes of the human gut microbiome.”nature 473.7346 (2011): 174-180.
9. Arumugam, Manimozhiyan, et al. Addendum “Enterotypes of the human gut microbiome.”nature 506.7489 (2014): 516–516.
10. Levy, Roie, and Elhanan Borenstein. “Metabolic modeling of species interaction in the human microbiome elucidates community-level assembly rules.”Proceedings of the National Academy of Sciences 110.31 (2013): 12804-12809
11. Fisher, Charles K., and Pankaj Mehta. “Identifying Keystone Species in the Human Gut Microbiome from Metagenomic Timeseries Using Sparse Linear Regression.” PLoS ONE 9.7 (2014).
Article provided by
Travis E. Gibson (@gibsonnews)
Harvard Medical School, USA
IFAC Technical Committee 1.2. Adaptive and Learning Systems