Reconceptualising Gut Microbiome Stability via Higher-Order Network Dynamics — Epoche C2
Research note: dissecting microbiome stability Date: 2023-10-27 The claim under examination is the one that opens a great many gut-microbiome papers: that a community with higher Shannon diversity is thereby more stable, and more resilient to perturbations such as a course of antibiotics or an abrupt change of diet. The index in question is $H = -\sum_{i=1}^{S} p_i \ln p_i$, where $S$ is the number of taxa and $p_i$ the proportional abundance of taxon $i$. This note argues that the claim fails, for a reason visible in the definition of $H$ itself, and that the network alternative usually offered in its place is at present asserted far more often than it is derived. What follows sets out what can be derived, and what it costs. The structural reason comes first, because it is decisive and takes one sentence. $H$ is a symmetric function of the abundance vector: permute the species labels and $H$ is unchanged. Any property of a community that depends on which species interacts with which is therefore invisible to it — not because the index is crude, but because it is defined on the wrong object. It is a functional of a distribution; stability is a property of a dynamical system. Nothing rules out an empirical correlation between the two, but a correlation is what it would be, and the mechanism would have to come from elsewhere. What May's criterion says, and why it applies to a gut community Start with the result that broke the older consensus: it is exact, and it points the opposite way to the folklore. Robert May, in 1972, considered a community of $S$ species at an equilibrium, linearised the dynamics about that equilibrium, and asked when the resulting community matrix $M$ — whose entry $M_{ij}$ is the partial derivative of species $i$'s growth rate with respect to species $j$'s density, evaluated at the equilibrium — has all its eigenvalues in the left half-plane, which is the condition for the equilibrium to be locally asymptotically stable. He gave the diagonal the value $-d$, representing self-limitation, and drew each off-diagonal entry independently: zero with probability $1 - C$, where $C$ is the connectance or the fraction of realised interactions, and otherwise from a distribution with mean zero and standard deviation $\sigma$. Write $M = -dI + A$. Each entry of $A$ has mean zero and variance $C\sigma^2$: with probability $C$ it contributes variance $\sigma^2$, and with probability $1-C$ it contributes nothing. The circular law for large random matrices states that the eigenvalues of an $S \times S$ matrix with independent, identically distributed, mean-zero entries of variance $s^2$ become uniformly distributed on a disc of radius $s\sqrt{S}$ centred at the origin. Substituting $s^2 = C\sigma^2$ gives a radius of $\sigma\sqrt{SC}$. Subtracting $dI$ translates the whole disc left by $d$, so every eigenvalue of $M$ has negative real part precisely when the disc clears the imaginary axis: $$\sigma\sqrt{SC} The exponent on $S$ is one-half, and not one, because it comes from the spectral radius of a random matrix rather than from any sum over species; this is the single feature of the result that does the ecological work. Fix $d = 1$ and $C = 0.1$. At $S = 200$ the criterion tolerates interaction strengths up to $\sigma = 1/\sqrt{20} = 0.224$. At $S = 800$ it tolerates $\sigma = 1/\sqrt{80} = 0.112$. Quadrupling the richness halves the admissible interaction strength, exactly because the radius grows as $\sqrt{S}$. Within this model, adding species destabilises. Two qualifications, since the result is routinely over-read. First, $M$ here is a random draw, so the conclusion is about the typical member of an ensemble and not about any particular assembled community; real communities are the survivors of an assembly process. Second, local asymptotic stability concerns the response to infinitesimal displacements from an equilibrium, which is not what a five-day course of ciprofloxacin delivers. Both are openings rather than refutations, and the first is the opening through which structure enters: if random matrices of a given $S$, $C$ and $\sigma$ are almost surely unstable, then the stable communities we observe must be non-random in some specific way, and the question is which way. Which structures buy stability Stefano Allesina and Si Tang answered that question in 2012 for the first non-random case that matters ecologically, and their result gives the phrase "how they interact, not how many there are" a precise meaning. Instead of drawing all off-diagonal entries independently, they draw them in pairs: the entries $M_{ij}$ and $M_{ji}$ describing the two directions of a single interaction are given a correlation $\rho$. The relevant limit theorem is then the elliptic law rather than the circular law: the eigenvalues fill an ellipse centred at $-d$ with horizontal semi-axis $\sigma\sqrt{SC}(1+\rho)$ and vertical semi-axis $\sigma\sqrt{SC}(1-\rho)$. Stability is governed by the rightmost point of the spectrum, so the criterion becomes $$\sigma\sqrt{SC}\,(1+\rho) The sign of $\rho$ is fixed by the type of interaction. In a predator–prey or host–parasite pair one entry is positive and the other negative, so $\rho 0$, the ellipse stretches, and tolerance falls. Interaction structure Pair correlation Criterion Tolerable interaction strength, relative to random Random pairing (May, 1972) $\rho = 0$ $\sigma\sqrt{SC} 1 Consumer–resource, sign-antisymmetric $\rho = -0.5$ $\sigma\sqrt{SC}(1+\rho) 2 Mutualism or competition $\rho = +0.5$ $\sigma\sqrt{SC}(1+\rho) 0.67 The ratios in the last column are $1/(1+\rho)$: at $\rho = -0.5$ that is $1/0.5 = 2$, and at $\rho = +0.5$ it is $1/1.5 = 0.67$. This is the sense in which network structure beats species count. The sign pattern of the interaction matrix enters the stability criterion as a multiplicative factor of order one, and $H$ cannot see it, since $H$ is a function of $p$ alone. Applying this to the gut requires care, so I state it as a condi