Research Article

Parametric transitions between bare and vegetated states in water-driven patterns

Matteo Bernard Bertagni, Paolo Perona, and Carlo Camporeale
PNAS August 7, 2018 115 (32) 8125-8130; first published July 23, 2018; https://doi.org/10.1073/pnas.1721765115

  1. Edited by Andrea Rinaldo, École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland, and approved June 26, 2018 (received for review December 14, 2017)

Significance

Since the appearance of land plants in Devonian time, vegetation has played a key role in the coevolution of life and landscapes as a result of mutual orchestrated processes between vegetation characteristics, environmental disturbances, and soil allometry. We mathematically frame the interactions between these three processes into a single parameter that discriminates between vegetated and bare states. In agreement with theories linking ecosystem development to hydrosphere and lithosphere connectivity, this theory suggests that the vegetation biodiversity of river sediment deposits occurs as a selection process in and among the biomechanical characteristics of species. Verified against field observations, this theory allows for water management applications and has important implications to understand natural and man-induced changes in biogeomorphological styles.

Abstract

Conditions for vegetation spreading and pattern formation are mathematically framed through an analysis encompassing three fundamental processes: flow stochasticity, vegetation dynamics, and sediment transport. Flow unsteadiness is included through Poisson stochastic processes whereby vegetation dynamics appears as a secondary instability, which is addressed by Floquet theory. Results show that the model captures the physical conditions heralding the transition between bare and vegetated fluvial states where the nonlinear formation and growth of finite alternate bars are accounted for by Center Manifold Projection. This paves the way to understand changes in biogeomorphological styles induced by man in the Anthropocene and of natural origin since the Paleozoic (Devonian plant hypothesis).

Mankind has lived by and controlled water systems (e.g., catchments, rivers, lakes, and sea environments) for millennia. Present day, about 80% of the world’s population lives close to and depends on freshwaters and related ecosystems. As a consequence, 48% of all rivers worldwide are hydrologically altered by human activities (1), with a biodiversity threat exceeding the 75th percentile (2). In the future, changes in precipitation-continental patterns due to global temperature rise over the 21st century (3) are expected to have tremendous impacts on freshwater biology [20,000 species are at risk (4)] by perturbing the ecogeomorphological equilibrium of fluvial landscapes and related water infrastructures. Although the important implications, which range from a better understanding of man-induced changes to studying the Devonian plant hypothesis (e.g., refs. 5 and 6), how and which physical conditions determine vegetation encroachment and pattern formation have never been fully analytically framed. This gap also limits our quantitative understanding of the impact of human activities and changing climatic scenarios on water-driven morphological patterns.

In a more physical parlance, surface-water bodies, such as streams, marsh lands, or coastal zones, behave as nonlinear dissipative dynamical systems characterized by stochastic processes, threshold mechanisms, and pattern formation induced by parametric transitions from different states. Such transitions can be smooth or critical [i.e., inducing catastrophic shifts, when small changes in certain parameters of a nonlinear system suddenly cause equilibria to appear or disappear (7⇓–9)].

Due to such a confluence of complexity, a quantitative physically based understanding of the relation between pattern formation and vegetation dynamics remains elusive, as in river systems for example. The reason for this failure is connected to our limited knowledge of the mutual triad interaction among three fundamental processes: (i) flow variability, (ii) sediment transport, and (iii) vegetation dynamics. These three cornerstones (henceforth referred to as ecomorphological triad) are common to many aquatic and aeolian morphogenic processes, and understanding their interactions is the mission of the emerging discipline of ecomorphodynamics (10). In practice, this goal has been addressed through field observations (11), experiments (12), or numerical simulations (13, 14). A limitation of these approaches is that they are partially able to shed light on the fundamental aspects of the involved dynamics and need thus to be complemented with analytical models. To this end, advances have been achieved by considering just two issues at a time, such as flow stochasticity and vegetation (15), vegetation and sediment transport (16).

In this article, a fully analytical physically based approach is formulated to link all of the three above mentioned aspects within the same theory and to address some open issues concerning vegetation inception and morphological pattern formation (17, 18). The rationale of the following theory is general, and it can be broadened to different kinds of spatially extended vegetated systems forced by time-dependent disturbances, such as salt marshes, wetlands, mangrove ecosystems, meandering rivers, aeolian dunes covered by biogenic crusts, or coastal foredunes (19⇓⇓–22) (see Fig. 1 A–C). Indeed, all these systems share very similar processes embedded into the above mentioned triad. Other remarkable examples of transitions—not considered here—are narrowing–widening and braiding–meandering (23). Instead, we apply our theory first to tackle a yet open puzzling question of physical geography: Why can two single-thread fluvial systems, having nearly the same hydrogeomorphologic features (sediment size d, slope S, and mean annual discharge Q¯), assume two completely different states, for example, fully vegetated or bare bars (see Fig. 1 D and E)? Physically speaking, the answer is that the phase space describing the dynamical system exhibits two different equilibrium points in the basin of attraction, thus making parametric transitions possible from one state to the other. We present an analytical theory that accounts for the aforementioned triad interaction and describes the vegetated–unvegetated transition in morphologically active rivers. Implications range from unraveling certain mechanisms underpinning effects of plants on rivers and landscape evolution through Earth history to understanding the role of environmental and anthropogenic disturbances in water-driven patterns under changing scenarios.

Fig. 1.
Fig. 1.

Examples of vegetation spreading in water-driven systems. (A) Porcupine River (Alaska). (B) Everglades wetlands (Florida). (C) Parabolic coastal foredunes (California). (D and E) Two straight rivers with alternate bars and very similar hydrogeomorphological conditions, flow from left to right. (D) Vegetated bars on Isère river near Arbin, France. (E) Bare bars on Alpine Rhine, near Vaduz, Liechtentstein. A, © 2018 Digital Globe.

Ecomorphological Triad

The main result of our theory states that vegetation encroachment on bare sediment undergoing occasional inundation is a secondary instability problem whose asymptotic behavior can be summarized into a single parameter, α—given by Eq. 6—which determines whether and how vegetation patterns develop. Thus, the spreading of plants is possible only if their biomechanical characteristics (e.g., growth rate and rooting efficiency) can cope with the removal action by stochastic floods in that particular sedimentary environment. The links among hydrology, biomechanics, and morphodynamics are all contained in the parameter α and determine its sign, which controls the transition from one state to another. Eventually, these processes select suitable plant characteristics in and among species and thus contribute to determine riparian vegetation biodiversity. We obtained this result by mathematically framing the processes of the ecomorphological triad as follows.

Flow Variability.

Modeling flow variability using a minimalist approach is a challenging task in river science, because of the nontrivial features of the discharge time series Q(t) (e.g., long-term correlation, intermittent behavior, etc.). The Compound Poisson Process (CPP) provides a parsimonious and robust strategy (24) by addressing the stochastic equation ∂tQ = F(t)−Q/τ, in which t is time, τ is the integral temporal scale (namely the integral of the autocorrelation function of Q), F is a shot noise with mean intervals between two pulses equal to τcv2 and mean value Q¯cv2, and Q¯ and cv are the mean and the variation coefficient of the discharge. A sample realization of CPP is reported in Fig. 2A (blue line). At steady state, the PDF of the CPP is Gamma-distributed (Fig. 2 A, Right) and the upcrossing time Tξ+—that is, the average time Q stays above a certain threshold ξ—is known in closed form (25)Tξ+=τeξ/Q¯cv2E1−1cv2,ξQ¯cv2,[1]where E(a,z) is the exponential integral function (26). Eq. 1 will be used later on in the triad interaction.

Fig. 2.
Fig. 2.

CPP (A and B) and alternate bar formation (C and D). (A) Simulated CPP for the river water discharge Q and its probability density function pQ (light blue lines). Inset shows the periodic signal used for the Floquet theory (red lines). (B) Water discharge series for the Isère and the Alpine Rhine (last 4 y reported). Notice the different shape of the PDF, with the sharper Gamma distribution for the Isère. (C) Saturation of bar amplitude during a time ts as described by the Landau Eq. 2. (D) Neutral stability curve discriminating the domain of bar formation (gray area) from the stable uniform solution (k is the longitudinal wavenumber, ds=0.005 and F=0.75).

Morphodynamics.

The second element of the triad interaction—sediment transport—is a threshold process that activates when the flow rate overcomes a lower critical value and becomes morphologically effective when an upper statistically rare flow rate Qf is overcome (Fig. 2A). This so-called formative discharge, an open issue in geomorphology, resets the riverbed by erasing any previous pattern and uprooting all vegetation. The bare flat state so created is the starting point of our analysis.

In a straight river with nonerodible banks, when Q>Qf, the sediment transport morphodynamically triggers the pattern formation of migrating two-dimensional bedforms, called alternate bars (Fig. 1 D and E), which were theoretically studied through linear and weakly nonlinear analyses (27, 28). Most previous studies neglected the role of flow variability, with the notable exceptions of refs. 29 and 30. Starting from infinitesimal perturbations, the bar amplitude A(t) asymptotically saturates to a finite value As in a nonlinear fashion, essentially following a Stuart–Landau equation (Fig. 2C):∂tA=ΩA−σ|A|A2.[2]For our convenience, we define the saturation time ts, such that A(ts) = 0.99As and A(0) = 0.01As, and we stipulate that a condition for bar formation is that the formative event must have a duration at least equal to the saturation time—that is, TQf+≥ts. An evaluation of the complex growth rate Ω and the first Landau coefficient σ is here given through the technique of the Center Manifold Projection (CMP), which, unlike the commonly used multiple scaling theory (28), offers a means to find As even away from the critical conditions but close to the neutral curve (31, 32).

Let us introduce the following scaling quantities: the river half-width B for the horizontal lengths, the depth D0 for the vertical length, U0 = Qf/2BD0 for the velocity, B/U0 for the time scale, and us = (Δ−1)gd for the sediment velocity (Δ, d, and g being the relative density, particle size, and gravitational acceleration, respectively). The governing dimensionless numbers result: the aspect ratio β = B/D0, the relative roughness ds = d/D0, the sediment transport capacity γ = dsus/(1−p)U0 with p the porosity of sediment, and the Froude number F = U0/gD0. With the above scaling and constant formative discharge Qf, momentum and mass conservation lead to the celebrated dispersion relation for bar linear instability (27) and to the neutral condition Ωr = 0 (subscript r refers to the real part). This condition manifests as a marginal curve in the longitudinal wavenumber–aspect ratio plane, with a minimum at the critical point (kc,βc) (see Fig. 2D). The linear analysis also provides the bar wavelength (fastest growing mode). Further details are provided in Methods and SI Appendix, Section S2.

The weakly nonlinear analysis through CMP leads to Eq. 2, from which one obtains As = Ωr/σr and ts∼6.6/Ωr. Furthermore, as the definition of volumetric discharge reads Qf = 2βFg(d/ds)5/2, both the saturation time and the formative discharge are functions solely of the morphodynamic parameters (ds, d, β, F). This allows us to define just two out of the three CPP parameters (Q¯,cv,τ), because the third one is given by the above mentioned condition for bar formation—that is, TQf+(Q¯,cv,τ) = ts. With the aid of Eq. 1, such condition leads toτ∼(tsQf)/(cv2Q¯), for Qf≫μ.[3]

Vegetation Dynamics.

After the formative event is extinguished, the flow decreases to ordinary values, reducing the water level and letting the crests of the recently formed bars emerge. At this point, vegetation may grow on the dried areas depending on flow variability. Vegetation is assumed to develop by following a logistic Holling type III equation, wherein the carrying capacity, K*, can be assumed as a quadratic function of the water depth, K* = K*(D) (15, 33). In fact, fluvial tree species usually perform the maximum growth when the water table is at an optimal depth: At lower stages, the roots cannot reach the water, while at higher stages, water logging occurs, thus reducing respiration and gas exchange in the root zone (34). In addition, the submerged sites experience a decline in vegetation because of anoxic conditions and uprooting induced by flow drag, which is proportional to the water velocity squared (16, 35). After assuming a linear relationship between flow drag and biomass removal and defining ψ as the density of biomass per unit area normalized to the maximum carrying capacity, we have∂ψ∂t=νgψ(K−ψ)−θ(D)νdD|U|2ψ,[4]where νg and νd are dimensionless growth and decay factors of vegetation and are proxies of the plant biomechanical properties in relation to the specific hydromorphological context and K is the normalized carrying capacity (see SI Appendix, Section S4); θ(⋅) is the Heaviside function (i.e., the decay is active only at submerged sites, when the water depth D is positive). We emphasize that, since the water velocity U = (U,V) and depth D account for the presence of the bar, the coefficients of Eq. 4 are space- and time-dependent. During ordinary flows, it is likely that sediment transport vanishes in most of the submerged sites, so that the bars can be approximately regarded as stable, until the occurrence of the next formative event. Moreover, the different time scales for bar saturation, order of days, and vegetation growth, order of years, allow us to separate mathematically the process and treat it analytically.

In this description, vegetation develops as a secondary instability (36) over the finite-amplitude bed topography. Encompassing bar topography in space and flow variability in time, a secondary instability is therefore performed by linearizing Eq. 4 about a new basic state (labeled with tilde), namelyD̃=Du(t)+AsD^1(k)f1(n)eiks.[5]In Eq. 5, the first term r.h.s. refers to the uniform solution, which is dependent on time through the CPP (Du(t) is related to Q(t) through Manning’s equation). The second term accounts for the presence of the finite amplitude bars (see Methods for more details), in which s and n are the longitudinal and transversal coordinates and D^1 and f1 are the linear perturbation of the water depth and its transversal structure. Similar forms are considered for the other variables. To make the computation analytically accessible, we substitute the discharge stochastic time series with a statistically equivalent periodic one, constituted by a repetition of the typical mean hydrograph. The latter is obtained by the following compatibility conditions (see red curve in Fig. 2A): (i) The duration of the event equals the average interval between subsequent shots, T=τcv2; (ii) the discharge mean value between formative events is preserved; and (iii) the peak value is such that the coefficient of variation cv of the stochastic series is preserved. In this manner, the time dependency in Eq. 4 becomes periodic and Floquet’s theorem can be applied. Thus, linearizing Eq. 4 around Eq. 5, the solution for the vegetation is ψ∼P(t)eα(s,n)t, where P(t) is a periodic function and α(s,n)∈R is the so-called Floquet exponent, which readsα=1T∫0T(νgK̃−νdθ(D̃)D̃|Ũ|2)dt,[6]where K̃ = K(D̃). We remind the reader that α is spatially distributed and its sign provides the asymptotic behavior of the secondary instability: Wherever α>0 vegetation patterns develop. The value of α is more easily evaluated numerically, as the analytical solution of the integral in Eq. 6 is particularly cumbersome. Although the above theory has been described for river systems, it is quite general, and the same steps could be repeated for other ecogeomorphologic systems driven by unsteady flows.

Results and Discussion

Parametric Transitions Driven by Flow Variability.

Once the triad interaction is set up (i.e., hydrodynamics, sediment, and vegetation parameters are provided), the present mathematical framework can evaluate the surface area of alternate bars where plants endure flow variability (α>0) between two formative events, resetting the morphology. A graphical example of finite amplitude computation and pattern formation is given in Fig. 3A. Let us define the AVI as the ratio of the vegetated area over the emerged area at the minimum discharge, Qmin—that is, the region theoretically colonizable by vegetation. Fig. 3B shows the dependence of AVI on flow variability cv, for two typical cases: a sand bed river (orange line) and a gravel bed one (light blue line). These two rivers share the same mean discharge Q¯, formative event 13Q¯, and channel width, but they differ in terms of the sediment and the Froude number F (a proxy of slope variation). This, in turn, affects the CPP correlation τ through Eq. 3. The two cases exhibit analogies and discrepancies. In fact, both rivers show that higher flow variability (increasing cv and thus Qmax) reduces the bar portion colonized by vegetation. In particular, there is a threshold value of flow variability, henceforth referred to as cv*, above which plant growth is completely inhibited (namely, AVI decays to 0). In such conditions, vegetation does not have sufficient time to develop since it is frequently removed by the flow. This cv* threshold corresponds to a parametric transition from vegetated to bare states. Concerning the discrepancies, the longer saturation time of the sand bed river (10 h) with respect to the gravel bed river (2 h) corresponds—via Eq. 3—to a discharge time series that is more correlated in the former case. This leads to longer submergence periods in the sand river and to plant uprooting for a fatigue stress. In the gravel river, the periods of bar submergence are instead shorter, but the more frequent events are equally able to remove mature vegetation. Another discrepancy concerns the two values of cv*, due to distinct slopes and bar heights. Under the same discharge conditions, the gravel bed river exhibits higher bars and shallower water depths and requires the maximum discharge Qmax (or flow variability) to be higher to flood the bar top and remove all of the vegetation.

Fig. 3.
Fig. 3.

Parametric transition driven by flow variability (all graphs obtained for {Q¯[m3/s],Qf,2B[m]} = {45,13Q¯,80}). (A) Density plot of α(s,n) and η(s,n) for a finite amplitude bed topography of a sand river {F,d[mm]} = {0.9,5}. The maximum (Qmax) and minimum (Qmin) water levels are marked by colored solid lines. (B) Areal Vegetation Index (AVI) versus cv for the sand bed river (orange line) and a gravel bed river (light blue line, {F,d[mm]} = {1.2,20}). In evidence, the threshold cv* discriminating between vegetated and bare state. (C) Transition variation coefficient cv* discriminating the bare (light brown) and vegetated state (green) as a function of the dimensionless vegetational parameters (same dataset of the sand river of b). As the root system strengthens, plants endure more easily flow variations (cv* increases). Dashed line is the prediction of Eq. 7. (D) The behavior of cv* in the parameter space. The triangles refer to the rivers of b. Gray and dashed areas correspond to no bar formation and no bedload, respectively.

The value of cv* is a key quantity, which results from an equilibrium between plant growth and decay. It corresponds to the flow variability that nullifies the integral (6) at the most favored site—that is, the bar top. The exact way to compute cv*, which is a cumbersome formulation, is provided in SI Appendix, Section S5. However, as the uprooting mechanism is much faster than plant growth, it is possible to simplify cv* evaluation assuming that no vegetational pattern develops when the maximum flow (Qmax) reaches the bar top. In this way, cv* can be numerically obtained as the solution of the following equation:D0(cv*,Qmax)+AsD^1(k)+D^1(−k)∼0.[7]

Fig. 3c shows that the above approximation is asymptotically exact for very fast uprooting (i.e., νd/νg≫1, typical values are of order 104). With the aid of Eq. 7, the behavior of cv* can be investigated in the parameter space (Fig. 3D). In the region where bedload transport occurs and alternate bars develop, cv* generally increases when moving from subcritical sand bed rivers (F<1, d<2 mm) to supercritical gravel bed (F>1, d>2 mm). This mechanism allows vegetation spread in mountain rivers with high flow variability.

Test Cases.

The computation of AVI is here shown for five actual fluvial cases of widespread interest to the scientific community (a summary of river features and validation results is provided in Table 1). The Isère (France) and the Alpine Rhine (border Austria–Switzerland) share almost identical hydrodynamic and sediment parameters (Fig. 1 d,e), but flood events of the latter are much stronger and flow variability is higher (37, 38). This implies slightly lower bar elevation in the Alpine Rhine, with frequent inundation and consequent inhibition of plant growth. In contrast, the slightly more regular discharges of the Isère favor vegetation development. Another test case concerns the Vedder canal in Canada (39). Upstream of the canal, the river is braided with multiple vegetated bars. In the canal itself, the bed is narrower with well-developed alternate bars that are easily submerged by ordinary flows. Thus, no vegetation develops in the canal (see SI Appendix, Fig. S5). Finally, the Arc river (France) has undergone a width reduction due to the construction of a highway in the 1990s (40). This triggered a parametric transition between a fully vegetated condition to a weakly vegetated bed (see SI Appendix, Fig. S6), where plants are periodically uprooted by annual flood events.

View this table:
Table 1.

Field data

As also reported in Fig. 4, the present theory quantitatively captures the underlying physics providing a good matching between theoretical predictions and measurements. A partial exception is the actual vegetation cover of the Arc river, which is not fully captured. This might be due to (i) the very strong anthropic influences upstream of the study area (dams and sediment mining) and (ii) the fact that the Arc river has not yet reached the ecomorphological equilibrium and thus we might expect bars to slowly vegetate in the future.

Fig. 4.
Fig. 4.

Comparison between theory (t) and measurements (m) for the bar wavelength L (blue), amplitude As (red), and AVI (green). Dataset is reported in Table 1.

These encouraging results may be useful to depict changes in biogeomorphological styles induced by man in the Anthropocene and of natural origin since the Paleozoic. For example, the model is able to capture the underlying physics for the river environments tested here and provide insights about the physical conditions that must subsist to allow for the colonization and the spreading of plants. Thus, the recent Devonian plant hypothesis, which suggests how plant root evolution might have contributed to the spreading of vascular plants (6), is supported by this theory. In fact, more robust root systems lead to a decrease in the uprooting coefficient νd and to a transition to vegetated states (Fig. 3C). However, further implications concerning how established plants have in turn affected landscape evolution are not contained in the present theory. This requires introducing the feedback of vegetation cover on morphodynamics and would explain why our river bar model underestimates the vegetated bar length (Table 1) with respect to the linear theory. Similarly, other processes not accounted for in this analytical theory concern sediment mining, alteration of sediment supply because of hydropower regulation, different root morphologies, and conditions for seed dispersal.

Conclusions

We presented a methodology to couple the three main processes controlling ecomorphodynamic pattern formation under a single analytical model. Results show that flow variability discourages vegetation growth, up to the point that above a certain threshold (cv* in Eq. 7) plant spread is completely inhibited. Such transition from vegetated to bare state depends parametrically on flow stochasticity, morphology, and plant biomechanical characteristics. In particular, cv* is generally higher for gravel-bed rivers than sand-bed rivers, and it increases for plants that can endure periodical submergence. This last outcome supports the idea behind the Devonian plant hypothesis, for which plant root evolution might have contributed to the spreading of vascular plants through the Paleozoic. The mathematical framework of this model can be adapted to other extended vegetated systems forced by unsteady flow conditions, and it may serve practitioners such as river scientists and water managers to predict the effect of changing flow-regime scenarios or better plan restoration works (41, 42).

Methods

Dimensionless shallow water equations for straight rivers (β≫F), under the quasi steady approximation (γ≪1), areU⋅∇U−1F2∇(D+η)−βCU|U|D=0,[8]∇⋅(UD)=0,[9]∂η∂t=γ∇⋅qs,[10]U⋅n^=qs⋅n^=0, (n=±1), [11]where ∇=(∂s,∂n), U=(U,V) is the depth-averaged flow field vector, C = C(D,|U|) is the friction factor, qs = qs(U,D,∇η) is the sediment bedload vector per unit width, n^ is the n axis unit vector, D is the water depth, and η is the bed elevation. The vector (U,V,D,η) contains the four unknowns of the PDE problem, Eqs. 8–11. Closure relationships for C and qs are reported in SI Appendix, Section S1. By perturbing the system around the basic uniform solution as (U,V,D,η) = (1,0,1,η0)+ϵu1 and substituting into Eqs. 8–10, one obtainsL0∂u1∂t=L1u1+ϵN(u1)+o(ϵ2) (ϵ≪1),[12]where L0,1 are (4×4) differential operators, whereas N(u1) only contains second-order nonlinearities. According to the approach of CMP, the unknown u1 follows the ansatzu1(s,n,t)=∑p=−∞+∞∑m=0nA[m,p](t)u^m(pk)fm(n)eipks,[13]where k is the longitudinal wave number and fm(n) are Fourier harmonics satisfying the boundary conditions (11), with m corresponding to the lateral Fourier order (m=1 is alternate bar mode, m=2 central bar mode, and so on). Finally, u^m are the eigenvectors of the classical linear theory (i.e., the case with N∼0 and A[m,p]=eΩt). Substituting Eq. 13 in the nonlinear system (Eq. 12) and taking the internal product with the adjoint eigenfunctions, a Galerkin system of nonlinear differential equations for the amplitudes A[m,p] can be obtained. The center manifold theorem (43) allows the amplitudes of the stable modes (fast stable manifold dynamics) to be projected on the neutral ones (slow center manifold dynamics). Thus, for any (m,p)>1, A[m,p] is written as a function of A[1,1]≡A and its complex conjugate, and Eq. 2 is eventually achieved. Further details on the derivation of the Landau Eq. 2 may be found in SI Appendix, Section S3, together with the analytical expressions of its coefficients Ω and σ.

Acknowledgments

We thank Guido Zolezzi for his useful advice on the validation and Alistair Borthwick for the editing.

Footnotes

  • ↵1To whom correspondence should be addressed. Email: matteo.bertagni{at}polito.it.

Published under the PNAS license.

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Parametric transitions between bare and vegetated states in water-driven patterns
Matteo Bernard Bertagni, Paolo Perona, Carlo Camporeale
Proceedings of the National Academy of Sciences Aug 2018, 115 (32) 8125-8130; DOI: 10.1073/pnas.1721765115
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