Correction for Clegg and Gross, Cross-feeding creates tipping points in microbiome diversity

Correction for “Cross-feeding creates tipping points in microbiome diversity,” by Tom Clegg and Thilo Gross, which published May 6, 2025; 10.1073/pnas.2425603122 (Proc. Natl. Acad. Sci. U.S.A. 122, e2425603122).

The authors note that the following equations in the published article appeared incorrectly:

On page 4, right column, 3rd full paragraph, line 5, the equation “R(x) = 1 + r(1 − x)” should instead appear as “R(x) = x + r(1 − x).”

On page 4, right column, 4th full paragraph, line 5, “y_m = (1 − s)z_m” should instead appear as “y_m = (s)z_m.”

The online version has been corrected.

The authors also note that the legend for Fig. 3 appeared incorrectly. They explain that the legend is missing some detail and should read “Points show solutions obtained from randomly generated networks with [mean degree z_c = z_m = 10.0,] N = M = 1,000 and 100 replicates at each value of s (or r below).”

The figure and its corrected legend appear below.

The online version has been corrected. The authors also note that in the SI Appendix, Eqs. 20–23 the proportions r and s should be inverted. As such, “A(x,r)” should instead appear as “A(x,1 − r)” and “A(x,1 − s)” should appear as “A(x,s)” in the equations.

Additionally, in SI Appendix, Eqs. 47 and 48 and the in-text equation in line 261 for the F(x) equations, the terms in the brackets should be flipped.

As such in line 261, the in-text equation “F(x) = exp ( λ (1 − x))” should instead appear as “F(x) = exp(z(x − 1)).”

For SI Appendix, Eq. 47, “G(x, y) = exp zⁱⁿ(1 − x) + z^out(1 − y) + r(1 − xy)” should instead appear as “G(x, y) = exp zⁱⁿ(x − 1) + z^out(y − 1) + r(xy − 1)”.

For SI Appendix, Eq. 48, “F(x) = F(x, 1) = exp (zⁱⁿ + r)(1 − x)” should instead appear as “Gⁱⁿ(x) = G(x, 1) = exp (zⁱⁿ + r)(x − 1).”

The SI Appendix has been corrected online.

The authors confirm that the typographical errors “do not alter the results, numerical simulations, or the broad interpretation” of this paper.

Fig. 3. Cultured community diversity collapses due to tipping points in cross-feeding networks. (A) Diagram showing how sampling a wild community and culturing on a given medium affects cross-feeding network structure in the model. From a wild community (1), sampling removes some proportion of consumer populations and any metabolite they produce (2). Supplying resources in culture allows the partial recovery of the cross-feeding network (3), resulting in the final community. (B) Plot showing how community diversity c∗ increases as a function of the intensity of sampling s. Lines show analytical solutions from Eq. 3, solid lines are stable and dashed unstable. Points show solutions obtained from randomly generated networks with [mean degree z_c = z_m = 10.0,] N = M = 1,000 and 100 replicates at each value of s (or r below). Dashed lines indicate the values of s in the plots below. (C) Plots showing the final community diversity c∗ against resource supply r across three levels of s. Again lines and points show solutions from analytical and numerical simulations respectively.

Supplementary Material

Appendix 01 (PDF)