Philippa Kaur

CORRESPONDENCE
22 June 2021

Ancient oaks of Europe are archives — protect them

Christian Sonne

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Changlei Xia

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Su Shiung Lam

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Christian Sonne

Aarhus University, Roskilde, Denmark.

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Changlei Xia

Nanjing Forestry University, Nanjing, China.

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Su Shiung Lam

University Malaysia Terengganu, Terengganu, Malaysia.

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Kongeegen, the King Oak, in Denmark could be up to 2,000 years old.Credit: Andreas Altenburger/Alamy

Some of the oldest trees in Europe are in danger because they are not being given the necessary level of protection. Oak trees (Quercus robur) that are more than 1,000 years old are found in the United Kingdom and in Fennoscandia, which includes Denmark, Sweden and Norway.For example, Denmark’s King Oak (pictured) is one of the world’s oldest living trees, dating to around 1,900 years of age. The United Kingdom has the largest collection of ancient oaks, reflecting 1,500 years of ship-building.The trees contain rings that represent archives of historical climate fluctuations and levels of atmospheric gases, so they can help to answer pressing questions about climate change and ecosystem dynamics (P. M. Kelly et al. Nature 340, 57–60; 1989).Fennoscandia and the United Kingdom could better safeguard their oaks using mechanisms such as those offered by the European Union’s Natura 2000 network of protected areas, or the protections conferred by UNESCO World Heritage sites in the United Kingdom. Otherwise, unsustainable management practices, deforestation, air pollution and climate change could leave these ancient species vulnerable to disease and extinction, with the loss of irreplaceable scientific information and cultural heritage.

Nature 594, 495 (2021)
doi: https://doi.org/10.1038/d41586-021-01699-0

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The authors declare no competing interests.

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Field location and soil samplingThe soil used in this experiment was collected from an agricultural field cultivated with maize at the “Instituto Tecnológico Superior del Oriente del Estado de Hidalgo” (ITESA) located in Apan, State of Hidalgo, Mexico (19° 73′ N, 98° 46′ W). The 0–20 cm top soil layer of three 400 m2 plots was sampled 20 times. The soil from each plot was pooled separately so that three soil samples (n = 3) were obtained. This field based replication was maintained in the greenhouse experiment so as to avoid pseudo-replication. The soil samples were passed separately through a 5 mm sieve and characterized.The soil is classified as a Phaeozem according to “World Reference Soil (WRS) system”, with pH 6.6, electrolytic conductivity (EC) 0.22 dS m−1 and water holding capacity (WHC) 515 g kg−1. The sandy clay loam soil with clay content 240 g kg−1, sand content 530 g kg−1 and silt content 230 g kg−1, had an ammonium content 8.16 mg kg−1 dry soil, nitrate 1.91 mg kg−1 dry soil and nitrite 0.01 mg kg−1 dry soil. The maize seeds were the hybrid variety 215 W obtained from Eagle® Sinaloa (Mexico).Characteristics of the biofertilizerAlthough a biofertilizer can be described in different ways we use the definition as given by38. Vessey defined (2003) a biofertilizer as “a substance which contains living micro-organisms which, when applied to seeds, plant surfaces, or soil, colonize the rhizosphere or the interior of the plant and promotes growth by increasing the supply or availability of primary nutrients to the host plant”. As the consortium used in this study fits the definition of a biofertilizer as given by Vessey38 we will refer to the consortium as the biofertilizer or when sterilized to the sterilized biofertilizer throughout the manuscript.The “biofertilizer” used in this study was a mixture of bacteria and leachate from compost of cow manure and was obtained from a local farmer in Hidalgo (Mexico) and characterized chemically and microbiologically. The cow manure was composted on a cement floor with a small inclination so that leachate could be collected easily. The farmer adds the leachate to the mixture of the bacteria to guarantee their survival and as an additional plant nutrient source. The farmer applies this solution regularly to fertilize his fields cultivated with maize. A same application protocol and procedure was used in this study to mimic the field experiment. Half of the biofertilizer obtained from the local farmer was sterilized by autoclaving at 121 °C for 20 min on three consecutive days so as to determine the effect of the microorganisms in the biofertilizer on the maize plants and the bacterial community structure, and the effect of the nutrients added with the biofertilizer.Experimental design and a greenhouse experimentThe research was done in a greenhouse at Cinvestav-Zacatenco situated to the north of Mexico City (Mexico). The experiment used a completely randomized block design with six treatments. The treatments combined as a first factor soil cultivated with maize or left uncultivated. A second factor included soil amended with the biofertilizer, sterilized biofertilizer or not fertilized. The daily temperature in the greenhouse ranged from 15 °C as minimum and reached a maximum 35 °C from April to August of 2017.As the experimental protocol was complex, a diagram of the different treatments and sampling is given in Supplementary Fig. S11 online. A total of 162 PVC columns with diameter 17 cm and height 60 cm were used in the experiment. Each pot was filled at the bottom with 0.5 kg tezontle, a highly porous volcanic rock, and 10 kg soil was added on top. The 162 columns included 6 treatments (uncultivated unamended soil, uncultivated soil amended with biofertilizer, uncultivated soil amended with sterile biofertilizer, maize cultivated unamended soil, maize cultivated soil amended with biofertilizer, maize cultivated soil amended with sterile biofertilizer; n = 6), 3 sampling times (day 44, day 89 and day 130; n = 3), three different soil samples (n = 3), with three columns planted with a maize plant per soil sample (n = 3). Three columns of each soil sample were planted with a maize plant to account for plants that might die so that at least one mature plant was obtained per treatment, sampling time and soil sample. The soil in the 162 PVC columns was adjusted to 40% WHC with distilled water and conditioned in the greenhouse for a week. Additionally, three PVC columns were filled with soil from each soil sample (n = 3), adjusted to 40% WHC with distilled water and conditioned for a week. These three soil samples were used to extract DNA as described below and defined the bacterial community at the onset of the experiment, i.e. time 0.Maize seeds variety 215 W Eagle hybrid seeds® were obtained from the farmer that provided us with the biofertilizer. Three washed maize seeds were planted at 3 cm depth in 81 columns, while the remaining columns were left uncultivated. Seven days after emergence, the most vigorous plantlet was kept and the other two discarded. After 44 days, the biofertilizer or the sterilized biofertilizer was diluted with water and applied with an atomizer (10 ml m−2 or similar to 100 l applied ha−1 by the farmer) so that it was added as fine spray evenly on soil of each pot when the seeds were planted. A similar volume of water was applied in the same way to the unfertilized treatment. Five more applications of the biofertilizer, sterilized biofertilizer or water by aspersion were done during the cultivation of the maize plants. As such, the uncultivated or maize plant cultivated soil was applied with the biofertilizer, sterile biofertilizer or water on 13th April, 28th May, 5th June, 13th July, 2nd August and 12th August 2017.Soil and plant samplingAfter 44 (27th May 2017), 89 (11th July 2017) and 130 days (21st August 2017), three columns from each treatment (n = 6) and soil sample (n = 3) were selected at random. Soil was removed from each column. The cultivated and uncultivated soil was sampled, characterized, and extracted for DNA as described below. The non-rhizosphere soil was separated from the rhizosphere soil by shaken the plants gently. The soil adhered to the roots was considered the rhizosphere soil. A 20 g sub-sample of the uncultivated, non-rhizosphere and rhizosphere soil was stored at − 20 °C pending extraction of DNA, while the pH and mineral N was determined in the remaining soil. Roots and shoots were separated, weighted and their length measured. The roots and shoots were dried in an oven at 60 °C for 24 h and weighed.Soil physicochemical characterizationThe moisture content of the soil was determined by weight loss after samples were dried at 60 °C in an oven for 24 h. The WHC was determined by saturating 50 g dry soil with distilled water, left to drain overnight and measuring the amount of water retained. The EC was measured in a soil paste (200 g soil/110 ml distilled H2O) with an HI 2300 microprocessor (HANNA Instruments, Woonsocket, RI, USA), while the particle size distribution was determined with the hydrometer method as described by Gee and Bauder39. The pH was determined in a 10 g soil–25 ml distilled water mixture with a calibrated pH meter (Denver Instrument, Bohemia, NY, USA) fitted with a glass electrode (3007281 pH/ATC Termofisher Scientific, Waltham, MA, USA).Mineral nitrogen (NO3−, NO2− and NH4+) was measured in the soil and biofertilizer. A 20 g soil sub-sample was extracted with 100 ml 0.5 M K2SO4 and filtered through Whatman filter paper® while mineral N was measured with a SKALAR automatic analyser system (Breda, the Netherlands)40. A 20 g biofertilizer sub-sample was mixed with 80 ml 0.5 M K2SO4, filtered through Whatman filter paper® and mineral N measured as described previously.DNA extraction and PCR amplificationA 5 ml sub-sample of the sterilized and unsterilized biofertilizer was centrifuged at 3500 rpm for 15 min and the supernatant removed. A 0.5 g sub-sample of soil was washed with 10 ml 0.15 mol l−1 sodium pyrophosphate to eliminate the humic and fulvic acids, centrifuged at 3500 rpm for 15 min and this process was repeated until the supernatant was clear41. The excess pyrophosphate was eliminated with 10 ml 0.15 mol l−1 phosphate buffer pH 8. Three different methods were used to extract DNA from the soil and the sterilized and unsterilized biofertilizer samples. The first technique was based on the method described by Green and Sambrook42. In the second method, cells were lysed with two lysis solutions and a thermal shock as described by Valenzuela-Encinas et al.43. The third method consisted of a mechanical disruption and detergent solution for cell lysis44. Each method was used to extract three times 0.5 g soil or 5 ml sterilized and unsterilized biofertilizer (a total of 1.5 g soil or 15 ml sterilized and unsterilized biofertilizer). The extracts from the soil and sterile or unsterilized biofertilizer were pooled separately.The 16S rRNA gene (V3–V4 region of bacteria) was amplified using the primers 341F (5′-CCTACGGGNGGCWGCAG-3′) and 805R (5′-ACHVGGGTATCTAATCC-3′45. The PCR conditions were 94 °C for 5 min, followed by 25 cycles of 60 s at 94 °C, 45 s at 53 °C, and 60 s at 72 °C, with a final extension of 10 min at 72 °C. The PCR was repeated three times for each sample. After PCR amplification, the obtained products were cleaned using the FastGen Gel/PCR extraction Kit (Nippon Genetics Duren, Germany) and quantified using a Nanodrop 3300 fluorospectrometer (TermoFisher, Wilmington, DE, USA) with PicoGreen dsDNA. The samples were mixed in equimolar amounts and sequenced using MiSeq 300-pb paired-end runs (Illumina, CA, USA) at Macrogen Inc. (Seoul, Korea).16S rDNA sequences analysisThe raw sequences were analysed with “Quantitative insights into microbial ecology pipeline” (QIIME) software (version 1.9.1)46. The barcode reads were demultiplexing removed from the sequences using the script extract_barcodes.py. The chimeric sequences were identified using “identify_chimeric_seqs.py” with the usearch61 method and removed47. The taxonomic assignment was done using the Ribosomal Data Project (rdp)48, against the Greengenes 16S rRNA database with a 0.8 confidence49. The sequences were clustered as operational taxonomic units (OTU) at 97% similarity level with the UCLUST algorithm47. Sequences were aligned against the Greengenes reference database using PyNAST version 1.2.250. The obtained 16S dataset was filtered, all OTUs assigned to Archaea were discarded and the dataset normalized. Alpha diversity indices (Chao1, Shannon and Simpson) were calculated from 478000 rarefied sequences with QIIME.Statistical analysisAll statistical analyses were done in R (R 4.0.2 GUI 1.72 Catalina build51). The characteristics of the maize plants (n = 3) obtained per plot (n = 3) were averaged and the sequences obtained from the replicate rhizosphere or non-rhizosphere soil were summed (n = 3) per plot before the statistical analysis. A non-parametric test was used to determine the effect of biofertilizer application and time on the plant and soil characteristics with the non-parametric t1way test of the WRS2 package (A collection of robust statistical methods)52. A non-parametric test was used to determine the effect of biofertilizer application or cultivation of maize on the bacterial alpha diversity with the non-parametric t1way test of the WRS2 package52. Heatmaps of the relative abundances of the bacterial groups were constructed with the pheatmap package53. Ordination [principal component analysis (PCA)], multivariate comparison (perMANOVA) and differential abundance (ALDEx2) was done with converted sequence data using the centred log-ratio transform test returned by the aldex.clr argument (ALDEx2 package54). The PCA was done with the vegan package55. Effect of biofertilizer application and cultivation of maize on the bacterial groups was determined using a compositional approach, i.e. analysis of differential abundance taking sample variation into account (aldex.kw argument, ALDEx2 package). A permutational multivariate analysis of variance (perMANOVA) analysis was also done with sequence counts converted using the centred log-ratio transform, i.e. aldex.clr argument (ALDEx2 package (aldex.clr(counts, mc.samples = 128, denom = ”all”, verbose = FALSE, useMC = FALSE)). The adonis2 argument (Vegan package) was used for the perMANOVA analysis to test the effect of cultivation of maize, time and its interaction, biofertilizer application, time and their interaction, and cultivation of maize, biofertilizer application and their interaction on the bacterial community structure (#adonis2(clrcounts ~ maize*biofertilizer, data = code, permutations = 999, method = ”euclidean”). Raw counts were used as input and Monte Carlo Dirichlet instances of the clr transformation values were generated with the function ‘aldex.clr’ of ALDEx2 (v.1.23.2) R package54. Distance pairwise matrices were calculated using the Aitchison distance and the principal coordinate analysis (PCoA) was calculated on the distance matrices with vegan R package55.Informed consentPermission was obtained from the farmer to use the maize seeds he provided.Ethical approvalThe experiment in the greenhouse complied with and was conducted as stipulated by national regulations. More

The Mara Hyena ProjectThis study uses data and samples from the Mara Hyena Project (approved by MSU IACUC and KWS), a long-term field study of individually known spotted hyenas that have been observed since May 1979. Study hyenas are monitored daily and behavioral, demographic, and ecological data are systematically collected and entered into a database. Here, we used data from four different hyena groups, called clans, as well as historic information about ecological conditions in the Masai Mara National Reserve. We maintained detailed records on the demographics of our study population, including sex, age, and the dates of key life-history milestones such as birth, weaning, dispersal and death. In the ensuing sections, we describe data collection and data processing procedures for assessment of T. gondii infection diagnosis, quantification of demographic and ecological determinants of infection status, and assessment of behavioral (boldness) and fitness (cause of mortality) characteristics hypothesized to be a consequence of positive T. gondii infection. The present analysis includes 168 hyenas, but specific subsamples vary depending on the particular hypothesis being tested.Biospecimen collection and assessment of Toxoplasma gondii exposureAs part of our long-term data collection, we routinely darted study animals in order to collect biological samples and morphological measurements. Of special relevance to this study is our blood collection procedure. We immobilized hyenas using 6.5 mg/kg of tiletamine-zolazepam (Telazol ®) in a pressurized dart fired from a CO2 powered rifle. We then drew blood from the jugular vein into sodium heparin-coated vacuum tubes. After the hyena was secured in a safe place to recover from the anesthesia, we took the samples back to camp where a portion of the collected blood was spun in a centrifuge at 1000 × g for 10 min to separate red and white blood cells from plasma. Plasma was aliquoted into multiple cryogenic vials. Immediately, the blood derivatives, including plasma, were flash frozen in liquid nitrogen where they remained until they were transported on dry ice to a −80 °C freezer in the U.S. All samples remained frozen until time of laboratory analysis for the T. gondii assays.Using archived plasma, we diagnosed individual hyenas using the multi-species ID Screen® Toxoplasmosis Indirect kit (IDVET, Montpellier). This ELISA-based assay tests for serological (IgG) reactivity to T. gondii’s P-30 antigen and has been used in many prior studies of T. gondii in diverse mammals22. The output of the assay is an SP ratio, which is calculated as colorimetric signal of immunoreactivity for a tested blood sample (S) divided by that of a positive control (P), after subtracting the background signal for the ELISA plate (i.e., a negative control) from both S and P. We tested 168 plasma samples from 168 individual spotted hyenas and determined infection status based on the kit manufacturer’s criteria for interpreting S/P: ≤ 40% = negative result, 40%  More

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Baseline model: no competitionWe start by assuming no competition and consider unconstrained growth in each of the two compartments. In this case, the equations describing our model become linear and can be rewritten in matrix form [4] as$$left( {begin{array}{*{20}{c}} {frac{{partial n_{H}}}{{partial t}}} \ {frac{{partial n_{E}}}{{partial t}}} end{array}} right) = underbrace{left( {begin{array}{*{20}{c}} {r_{H} – m_{E}} & {m_{H}} \ {m_{E}} & {r_{E} – m_{H}} end{array}} right)}_{{mathrm{projection}}, {mathrm{matrix}}}left( {begin{array}{*{20}{c}} {n_{H}} \ {n_{E}}end{array}} right)$$
(2)
The dominant eigenvalue λ of the above-defined projection matrix gives the asymptotic overall growth rate of the considered microbial lineage. This quantity is an appropriate measure of fitness [4] insofar as it measures reproductive as well as transmission success and recapitulates the effects of all the life-history traits (rE, rH, mE, and mH, also defining the phenotype in our model). Overall microbial fitness is thus integrated across the different steps of the life cycle, thereby considering the reproductive rates (i.e., replication rates) within each of the compartments and importantly transmission rates (i.e., migration rates) across the compartments. The dominant right eigenvector represents the stable distribution of microbes in the two compartments, and the number of microbes in each of the compartments grows exponentially with rate λ. The value of λ can be calculated at each point of the phenotypic space defined by the ranges of possible values that could be taken by the life-history traits rE, rH, mE, and mH. The dependence of λ on these traits tells us at which points of the phenotypic space fitness is maximized and how it can be increased at all other points.From the projection matrix, we calculate the dominant eigenvalue as$$lambda = frac{1}{2}left(sqrt {left( {r_E + r_H – m_E – m_H} right)^2 {,}-{,} 4left( {r_Er_H – r_Em_E – r_Hm_H} right)} + r_E +r_H – m_E – m_H right).$$
(3)
Note that if microbes replicate at the same rate in the host and in the environment, i.e., if rE = rH = r, λ simplifies to r, regardless of the migration rates mH and mE. When there is an asymmetry between the two replication rates however, which is very likely to be the case in nature, then the migration rates also affect the overall growth rate. In the following sections, we study this effect compared to the effect of the replication rates. We arbitrarily set rH ≤ rE, and rE  > 0 – otherwise the lineage goes extinct. In biological terms, this corresponds to the situation where the microbial lineage is initially more adapted to the environment than to the host and thus grows faster in the environment. But mathematically, in this model, host and environment are symmetrical, i.e., they only differ by the rates defined above. Thus, the chosen direction of this inequality does not carry any strong meaning, and there is no loss of generality in making this choice. In particular, one can access the opposite biological situation where microbes replicate faster in the host than in the environment – as is the case for viruses, that can only replicate in the host (rH  > 0) but decay in the environment (rE  0. Setting rE = 1 to scale time (and thus, measuring all other rates in units of the replication rate of the microbe in the environment), λ reduces to$${uplambda}_{sym} = frac{1}{2}left( {1 + r_H – 2m + sqrt {left( {1 – r_H} right)^2 {,}+ {,}4m^2} } right)$$
(4)
For any fixed positive value of m, λsym is a strictly increasing function of rH, which reflects the fact that increasing rH allows for additional growth within the host. We will limit ourselves to the study of rH ≥ −1, which ensures a positive value for λsym. For any fixed value of rH, λsym is a decreasing function of m, which reflects the fact that for increasing m, microbes are increasingly lost towards the host, where growth is slower than in the environment. Figure 1C shows the value of λsym on the reduced phenotypic space defined by rH and m. The maximum possible value for λ is 1 (in units of rE). This value is achieved either by increasing the ratio of replication rates between host and environment, so that the replication rates in both compartments are identical (strategy I), or by reducing migration between host and environment, and in particular, by reducing mH (strategy II). This second strategy allows microbes to spend a longer time in the environment on average. Note however, that this strategy is limited, since setting m to zero decouples the two compartments completely, in which case the microbial lineage is no longer subject to a multi-step life cycle.How strong is the selection on these traits? This question can be approached by inferring how strongly the overall growth rate depends on the traits we are considering. One standard approach to measure this is sensitivity analysis [4]. One defines the sensitivity of the overall growth rate λ achieved by the phenotype described by the vector x = (x1,…, xN) in the trait space to its ith life-history trait as$$s_{mathrm{i}}left( {mathbf{x}} right) = left. {frac{{partial {uplambda}}}{{partial {mathrm{x}}_{mathrm{i}}}}} right|_{mathbf{x}}$$
(5)
This quantity gives the change in the value of λ that results from a small increment of the trait i. It is a local property that can be calculated for each point ({mathbf{x}}) of the trait space. The vector of the sensitivities at point ({mathbf{x}}) gives the direction of the selection gradient on the fitness landscape. In other words, to achieve efficient phenotypic adaptation, the lineage should move in the trait space following the direction of this gradient.If the lineage can invest in phenotypic adaptation only by tuning one of its life-history traits at a time, then it should act upon the trait that has the largest (absolute) sensitivity at the current position of the lineage in the trait space. In our model, in all generic cases (i.e., when m  > 0), the largest sensitivity is always associated to the increase of the trait rE, the replication rate in the fast-growing compartment. However, we assume that the considered microbial lineage is initially fully adapted to the environment, so that it has reached its evolutionary limit, and we can essentially ignore the sensitivity to rE throughout the manuscript to focus on the sensitivity to the other traits. This reasoning allows to divide the trait space into regions of distinct optimal strategies, as shown in Fig. 1C. In the regime of high migration rates (i.e., when the switch between the compartments is so rapid that the microbial lineage is almost experiencing a habitat having average properties between the host and the environment), strategy I (increasing rH) becomes almost always optimal, except for small replication ratios, where there is almost no replication in the host. In summary, migration rates are important when replication in the host is slow compared to the environment, and when migration itself is slow. These conclusions remain qualitatively unchanged with asymmetric migration rates, although a third optimal strategy (increasing mE) appears for an intermediate region of the traits space when the asymmetry is important (see electronic Supplementary Material (ESM) section 1 and Supplementary Fig. S1).Model with global competition between all microbesIn the baseline model, there are no constraints on growth. In nature, however, microbes do face limits to their growth. Since the equations above are linear and can only give rise to exponential growth or exponential decay, they can only describe the microbial dynamics over a limited period of time. In order to account for saturation and competition during growth, we thus need to introduce non-linear terms to the equations (1). The study of this kind of systems often focus on long-term dynamics, yet it can be of high practical relevance to study the transient optimal strategies, as shorter timescales are often relevant in the real world – whether it be due to experimental constraints or to ecological disturbances and perturbations [20]. Since we are going to consider some out-of equilibrium dynamics, in particular in the section with competition limited to one of the compartments, and because we are also interested in transient properties, we will adopt a numerical approach based on the number of microbes [21, 22].In this section, we study the case of a microbial lineage constrained by global competition occurring at rate k = kHH = kEE = kEH = kHE. This situation could correspond to a host-associated microbe living in direct contact with an external environment, e.g., on the surface of an organism. Alternatively, what we call the “environment” in our model could represent another host compartment in direct contact with the other, like the gut lumen and the colonic crypts. In that case, microbes living in association with the host are in direct contact with those in the environment and can mutually impact each other’s growth. This is of particular relevance if microbes living in both compartments rely on and are limited by the same nutrients for growth.From the microbial abundances in the two compartments obtained by numerically solving the equations, one can build a proxy for the overall growth rate of the microbial lineage. To remain consistent with the previous section, we define$$varLambda left( {mathbf{x}} right) = frac{1}{{t_{max}}}log left( {frac{{n_Eleft( {t_{max}} right) + n_Hleft( {t_{max}} right)}}{{n_Eleft( 0 right) + n_Hleft( 0 right)}}} right)$$
(6)
i.e., the effective exponential growth rate of the microbial lineage over a chosen period of time [0, tmax]. Figure 2A provides a graphical explanation for the expression of Λ. There are indeed several fundamental differences between the effective exponential growth rate Λ in a non-linear system and the asymptotic growth rate λ in a linear system, the dominant eigenvalue of the projection matrix as defined in the baseline model. First, Λ provides a measure of growth for the whole lineage, but is not an asymptotic growth rate (as compared to λ in the baseline model): in the case of global saturation, replication stops when the carrying capacity is reached, and the asymptotic growth rate for the whole lineage would thus be zero. Therefore, the choice of the probing time tmax has an impact on Λ, as shown in Fig. 2A. Second, the choice of the exact form of Λ now implies biological assumptions on the selection pressure experienced by the microbial lineage: choosing the effective exponential growth rate over the whole lineage as we do implies that selection is acting on both compartments evenly. There may be some situations in which the microbes in one of the compartments only are artificially selected for (e.g., as part of the protocol of an evolution experiment). In such cases, it would make sense to define Λ as the effective exponential growth rate over just this compartment. This may lead to different conclusions, in particular at the transient scale. One must thus adapt Λ to the specifics of the modeled system. In addition, the choice of tmax itself has a biological meaning, and should in particular not exceed the time upon which the dynamics of the system are accurately described by the set of equations. This may also be determined by experimental times.Fig. 2: Optimal strategies in the model with global competition.A Temporal dynamics of the total number of microbes nE(t) + nH(t) for three different sets of traits values, differing only by their intensity of competition k = kHH = kEE = kEH = kHE. Other parameter values are: rH = 0.1, mE = mH = 0.5. The effective overall growth rate Λ is calculated numerically by taking the slope of the straight line that connects the abundances in t = 0 and in tmax, thus making Λ a quantity that strongly depends on tmax. B Change in the contour line delimiting the regions of optimality of the two optimal strategies (strategy I: increasing rH; strategy II: decreasing mH) with tmax, the time chosen to measure the final number of microbes, measured in units of 1/rE. Initially the microbes are equally distributed between the host and the environment. Supplementary Fig. S2 shows how this is modified with different initial conditions. Because in this model all the microbes are equally impacted by competition, with tmax large enough, one recovers the contour line of the baseline model calculated analytically (black line). Continuous lines: k = 0, i.e., no competition. Dashed lines: increasing values of k (competition intensity). C, D Change in the fitness landscape with tmax (panel C: tmax = 0.7 and panel D: tmax = 3). The colored lines show the contour delimiting the regions of optimality of strategies I and II for three different values of k, as shown on panel B. Black line: long-term limit of no competition from the base model.Full size imageWe now calculate the sensitivity of Λ in the direction of the trait i at the point x of the phenotypic space as$$S_i = frac{{varLambda left( {x_1,x_2, ldots ,x_{i – 1},x_i + delta x_i,x_{i + 1}, ldots ,x_N} right) – varLambda left( {x_1,x_2, ldots ,x_N} right)}}{{delta x_i}}$$
(7)
with δxi the discretization interval, and N the number of traits defining a phenotype x.For this numerical approach, additional choices need to be made. First, the trait space needs to be discretized. Then, to calculate Eq. (7), one needs to choose a set of initial conditions and a probing time at which to measure the microbial abundances, as exposed in detail for the linear case in [20]. Finally, we need to choose the discretization interval δxi. In the following, we always choose δxi sufficiently small for convergence, i.e., so that it does not significantly impact the numerical values of the sensitivities, and focus on the choices of the other parameters (probing time and initial conditions) and the influence of the competition intensity k. One strategy to explore the possible impact of initial conditions is to use “stage biased vectors” [20], i.e., extreme initial distributions of microbes across the two compartments. This corresponds to initial conditions where microbes either exist only in the host or only in the environment.In Fig. 2B, we show how the contour lines delimiting the two optimal strategies change with the final time tmax chosen to measure the overall growth rate and with the intensity of competition k, for a mixed initial condition (nE(0) = 0.5, nH(0) = 0.5), and Supplementary Fig. S2 shows how this is modified with stage biased vectors. In all cases, with sufficiently long tmax, the contours converge to the contour plot of the baseline model shown in the previous section. This is expected, since competition here affects all the microbes in the same way, so that the equilibrium distribution is the same as the asymptotic distribution of the baseline model (given by the dominant eigenvector). Mathematically, global competition can be seen as a modification of the baseline projection matrix by subtracting an identity matrix times a scalar depending on time. This does neither affect the eigenvectors nor the dependence of the dominant eigenvalue on the traits.In the case where all the microbes are initially in the environment (Supplementary Fig. S2A), there is no transient effect and whichever tmax is chosen, all the contour lines collapse to the limit of the baseline case. In the case where all the microbes are initially in the host (Supplementary Fig. S2B), a third optimal strategy transiently appears (increasing mE) and remains at long times around m = 0. In this unfavorable condition (m = 0 and an initially empty environment), increasing the microbial flux towards the environment becomes more important than limiting the flux of microbes leaving it (which is nonexistent when m = 0).Finally, we observe that the intensity of competition has only a small effect on the contours (Fig. 2B and S2B), but increasing k appears to slightly accelerate convergence to the baseline contour. By limiting growth in the host compartment – when it is initially relatively more populated than in the asymptotic distribution – competition facilitates the convergence to the baseline asymptotic distribution, where most of the microbes live in the environment.Model with competition within one of the compartments onlyIn this section we consider competition happening inside one of the compartments only (i.e., kEH = kHE = 0 and kEE ≠ 0 or kHH ≠ 0). We will start by considering competition in the host only (the slow-replicating compartment). In a second step we also look at the case with competition limited to the environment. One should bear in mind that it also covers the case of competition limited to a host where replication is faster than in the environment (rH  > rE), provided a switch of the H and E index.In the case where competition is limited to only one of the compartments, we do not expect an equilibrium to exist for all traits combination of the phenotypic space. If migration is not sufficiently important, the number of microbes in the unconstrained compartment keeps increasing exponentially faster than the number of microbes in the constrained compartment, which contribution to the whole lineage thus becomes rapidly negligible. At sufficiently high migration rates however, an equilibrium is expected, because microbes switch habitats sufficiently rapidly for competition to be globally effective, although it directly affects only one of the compartments.Competition in the host only (slow-replicating compartment)When there is competition in the host only, there is no (positive) equilibrium for all mH  More

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