A Marrocco

H Henry

I B Holland

M Plapp

S J Séror

B Perthame
email: benoit.perthame@upmc.fr

Models of self-organizing bacterial communities and comparisons with experimental observations

Keywords: Dendritic patterns, Bacillus subtilis swarming, Reaction-diffusion equations, Cell community growth. AMS subject classification: 35K55, 65M60, 92C17

Bacillus subtilis swarms rapidly over the surface of a synthetic medium creating remarkable hyperbranched dendritic communities. Models to reproduce such effects have been proposed under the form of parabolic Partial Differential Equations representing the dynamics of the active cells (both motile and multiplying), the passive cells (non-motile and non-growing) and nutrient concentration. We test the numerical behavior of such models and compare them to relevant experimental data together with a critical analysis of the validity of the model based on recent observations of the swarming bacteria which show that nutrients are not limitating but distinct subpopulations growing at different rates are likely present.

Introduction

Communities of cells can exhibit remarkable patterns which have attracted the attention of scientists for many years. They result from highly complex but poorly understood interactions between cells and internal regulatory networks, which involve both chemical signaling and the effects of physical factors. Numerous models have been used in attempts to represent some of these biophysico-chemical effects and to describe the resulting self-organizing patterns.

Different biophysical factors involved in pattern formation have given rise to various types of modelling. One class of model concerns auto-chemotaxis (attraction of cells by a chemical substance emitted by the cells themselves) and gives rise to a Fokker-Planck equation that is commonly called the Keller-Segel system after the seminal work [START_REF] Keller | Model for chemotaxis[END_REF]. This model is mathematically very challenging and has motivated numerous studies (see [START_REF] Blanchet | Two dimensional Keller-Segel model in R 2 : optimal critical mass and qualitative properties of the solution[END_REF][START_REF] Perthame | Transport Equations in Biology[END_REF] and the references therein); in particular this kind of model typically leads to cell aggregation in one or several discrete spots (blow-up of the system as a Dirac mass solution). Several such models and numerical results are presented in [22].

Other models are based on the multiplication of cells resulting from nutrients initially present in the medium and consumed by the expanding community, combined with active and random motion of bacteria. This approach has also been widely used and can generate dendritic patterns in the absence of oriented drift (preferred direction of motion) in contrast to the Keller-Segel model which describes cells moving preferentially towards higher concentrations of the chemo-attractant. Many additional factors have been incorporated into models, such as the observed higher motility of cells at the tip of the dendrites (region of higher population density and higher nutrient concentration) in [START_REF] Kawasaki | Modeling spatiotemporal patterns created by Bacillus-subtilis[END_REF], a surfactant secreted by the cells that may change the liquid surface and thus the migration speed of cells [START_REF] Kozlovsky | Lubricating bacteria model for branching growth of bacterial colony[END_REF][START_REF] Golding | Studies of bacterial branching growth using reaction-diffusion models for colonial development[END_REF], or differentiation from swimmers to swarmers for Proteus mirabilis as modelled in [START_REF] Esipov | Kinetic model of Proteus mirabilis swarm colony development[END_REF][START_REF] Frénod | Existence result of a model of Proteus mirabilis swarm[END_REF].

We are interested in models that can explain the dendritic patterns exhibited by swarming communities of Bacillus subtilis, taking into account possible biophysical effects arising during the migration. We review several such models and show numerical simulations (Section 2.), summarize some recent experimental observations in Section 3., and discuss in Section 4. the similarities and differences between simulations and experimental results.

Models for self-organizing communities

The mathematical description is performed in two steps. First, a model is proposed based on the major supposed mechanisms that drive the migration of the cells. We give two such examples: one is based on nutrient consumption, the other on chemoattraction and chemorepulsion. Secondly, a numerical simulation is performed that allows us to visualize the approximate solution.

Throughout this section we consider a general situation where the cell community lives in a domain Ω. For numerical simulations we consider only the case of a disk that represents the usual experimental domain (a Petri dish). The initial state, mimicking the experimental device, is set with an initial value for the bacterial population density representing an 'inoculum' in the center of the disk.

Reaction-diffusion models (nutrient only)

The nutrient-based models used to generate dendritic patterns of cell communities are mostly formulated in terms of three quantities

• the population density u(x, t) of active cells at the location x ∈ Ω. Under the effect of their flagella, active cells undergo a random movement resulting in a diffusion of intensity d u , and they multiply according to the nutrient available locally • the nutrient concentration v(x, t) diffuses according to Einstein's rule and, because the nutrient is limited, it can diminish locally due to its consumption by multiplying cells • the population density of passive cells w(x, t). For these cells the effect of their motion and multiplication is neglected. Active cells become passive according to some rules that differ from one model to the other, and then they stay passive, i.e. they do not move or multiply.

These ingredients lead to write general systems of the form

             ∂ ∂t u(x, t) -d u ∆u(x, t) = u[vf (u, v) -g(u, v)], ∂ ∂t v(x, t) -d v ∆v(x, t) = -uvf (u, v), ∂ ∂t w(x, t) = ug(u, v).
(2.1)

Figure 1: Dynamics of community growth as given by a numerical simulation of the Gray-Scott model (2.2) with n = 1. Active cells modelled by the component u(x, t) are shown (total number of cell presented in Fig. 2). Three different successive times are presented. These cells are positioning where nutrient is available.

Such systems have been introduced to model chemical reactions, and the Gray-Scott system [START_REF] Gray | Autocatalytic reactions in the isothermal continuous stirred tank reactor: isolas and other forms of multistability[END_REF] is a simple and classical example which writes as

             ∂ ∂t u(x, t) -d u ∆u(x, t) = u[u n v -µ], ∂ ∂t v(x, t) -d v ∆v(x, t) = -u n+1 v, ∂ ∂t w(x, t) = µu(x, t).
(2.2)

Figure 2: Same model as in Figure 1, the total population density u + w is shown. The dendritic pattern is determined by the passive cells in this type of model.

Figure 3: Same model as in Figure 1, the nutrient concentration v is shown (dark represents high concentration). While moving outwards towards undepleted nutrient, the active bacteria consume the nutrient.

Here n ≥ 0 is an integer related to the mass action law for the molecules undergoing the chemical reaction and µ > 0. Variations around this model can also be interpreted in terms of bacterial motion as proposed in Kessler and Levine [START_REF] Kessler | Fluctuation induced diffusive instabilities[END_REF], Golding et al [START_REF] Golding | Studies of bacterial branching growth using reaction-diffusion models for colonial development[END_REF]; they replace the growth term u n v by h(u)v where h(•) is a truncation function for small values of u and h ≈ 1 for large values.

The Gray-Scott model explains the instability that generates the digitation process. It is related to concentration effects of the equation on active cells; its solution u exhibits high values on the tip of the dendrite and move outwards where nutrients are replete (Figure 1). These concentration points are traveling pulses that undergo secondary instabilities which explain their branching, see [START_REF] Muratov | Traveling spike autosolitons in the Gray-Scott model[END_REF][START_REF] Kolokolnikov | The existence and stability of spike equilibria in the onedimensional Gray-Scott model: the pulse-splitting regime[END_REF]. They leave behind them the column of passive bacteria forming the dendritic pattern shown in Figure 2. In order to run simulations on the normalized unit disk, we have used parameters given by

n = 1, d u = 6.25 10 -8 , d v = 100d u , µ = 0.01, v 0 = 1,
where v 0 is the initial constant value of v.

Rather than a limitation of growth for small values of u as in the Kessler and Levine model, Mimura et al [START_REF] Mimura | Reaction diffusion modelling of bacterial colony patterns[END_REF] proposed a limitation on the transition rate to the passive state for large values of u or v. The choice of the reaction terms f and g in the general system (2.1) is then given by

               ∂ ∂t u(x, t) -d u ∆u(x, t) = u v - µ (a + u)(b + v) , ∂ ∂t v(x, t) -d v ∆v(x, t) = -uv, ∂ ∂t w(x, t) = µu (a + u)(b + v) . (2.3)
The resulting dendritic patterns differ slightly from those obtained with the Gray-Scott model. But the underlying mechanism is very similar, as can be seen in Figure 4 which shows the active cells and the total population. The simulations are run in the unit disk and the parameters we have used for model (2.3) are

d u = 1.4 10 -7 , d v = 20 d u , a = 1/2400, b = 1/120, µ = ab, v 0 = 0.087.
Other biophysical processes can be taken into account in this kind of models in order to explain the behaviour of specific bacterial communities. The effect of surfactant has, for instance, been analysed in [START_REF] Kozlovsky | Lubricating bacteria model for branching growth of bacterial colony[END_REF] and affects the diffusion term by allowing higher motility of cells depending on the height of the surface liquid. The influence of the reaction terms and analogy with phase transitions are studied in [START_REF] Wakano | Phase transition of traveling waves in bacterial colony pattern[END_REF]. The references [START_REF] Golding | Studies of bacterial branching growth using reaction-diffusion models for colonial development[END_REF]22] also contain several other models.

Chemoattraction, chemorepulsion and Fokker-Planck terms

When the bacteria emit a chemoattractant or chemorepellent substance, this results in additional terms in the reaction diffusion systems described above. Assuming that the medium is rich enough and therefore the nutrient is not limiting, we arrive at systems of the form

                     ∂ ∂t u(x, t) -d u ∆u(x, t) + div[u(∇c a -∇c r )] = u[f (u) -g(u, c r )], ∂ ∂t w(x, t) = ug(u, c r ), ∂ ∂t c a (x, t) -d a ∆c a (x, t) + τ a c a = ρ a h a (u, w), ∂ ∂t c r (x, t) -d r ∆c r (x, t) + τ r c r = ρ r h r (u, w). (2.4)
Here c a and c r represent the concentration of chemoattractant and chemorepellent. These are assumed to diffuse according to Einstein's rule with coefficients d a and d r , they are degraded with the rates τ a and τ r (depending possibly on the cell population densities u and w), and they are secreted by the cells with rates ρ a and ρ r . Their actions are represented by Fokker-Planck terms in the equation for u, together with the Keller-Segel model mentioned earlier.

In this combination of reaction-diffusion models together with drift terms, the latter represent chemoattraction/chemorepulsion and have tendency to dominate the dynamics. This yields much more dynamical profiles and stronger aggregation effects on active cells as can be seen in Figure 5 where we have assumed a short-range attraction on active cells (u component) and long-range repulsion on passive cells (w component).

Numerical method

The numerical solutions we have presented are based on the mixed finite element method (with Raviart-Thomas elements of lowest degree often denoted by RT 0 ) that has been described in [START_REF] Marrocco | 2D simulation of chemotactic bacteria aggregation[END_REF][START_REF] Marrocco | Simulations numériques de modèles de réaction-diffusion à l'aide d'éléments finis mixtes[END_REF]. One of the major ingredients is a change of unknown functions to handle the several decades in population densities that can occur in different areas of the computational domain. This is run on an unstructured triangular grid as shown in Figure 6 which allows the instabilities to progress without preferred direction, in contrast to rectangular grids which have the tendency to create mesh effects.

For numerical purposes the domain is the unit disk (or a sector of it) and the diffusion coefficients are small. This forces us to use fine grids (finer as the diffusion coefficients become smaller).

The dendritic patterns are very dependent on the grid used for computing and convergence is reached with very fine grids. For this reason the numerical convergence has only been reached for relatively moderate values of d u and of the ratio d v /d u .

Experimental results

Bacillus subtilis is a non-pathogenic but important constituent of soil and the plant rhizosphere. This bacterial species is also one of the major model organisms used in the laboratory throughout the world to study fundamental questions concerning bacterial growth, metabolism, physiology and behaviour. B. subtilis is now an important model for studying the life style and social behaviour of bacteria as large communities -the normal form of most bacteria in nature. A particularly remarkable form of such community growth is the ability of B. subtilis to 'swarm' over the surface of low concentration agar (0.7%-1%). Swarming is a process of rapid mass migration of cells over a surface, involving a co-operative interaction between cells but not necessarily involving cell aggregation. We are studying the swarming of strains of B. subtilis over a fully defined medium (B-medium) in a Petri dish (a swarm plate), in which the bacteria migrate from a central inoculum as hyper-branching dendrites, forming radiating patterns covering several square centimeters in a few hours ( [START_REF] Julkowska | Branched swarming patterns on a synthetic medium formed by wild type Bacillus subtilis strain 3610[END_REF][START_REF] Julkowska | Comparative analysis of the development of swarming communities Bacillus subtilis 168 and a natural wild type: critical effects of surfactin and the composition of the medium[END_REF]).

The presence of flagella and the secretion of a surfactant (surfactin) by the bacteria, plus the products of at least 15 genes, are absolutely essential for swarming. Following inoculation of the plate with 10 4 cells, the bacteria multiply with an estimated generation time (doubling time) of about 90 min, compared to 110 min when cells are growing in a classical shaking liquid culture. After 11-12 h of growth, the inoculum forms the mother colony (MC), approximately 30 µm thick, 2 mm in diameter. This growth period is presumably necessary to build a critical mass and an accumulation of a chemical signal sufficient to trigger in some cells the ability to form dendrites. The first visible sign of initiation of swarming is the spreading outwards from the edge of the MC of a transparent zone of surfactin. Approximately one hour later, hemispherical 'buds' approximately 800 µm in diameter, abruptly appear (burst) from the edge of the MC. These form the heads (tips) of the rapidly elongating 10-14 primary dendrites. Surfactin production is essential for formation of the pre-dendrite buds and experiments suggest that its presence modifies the surface of the agar gel, presumably by inducing the formation of a thin layer of fluid close to the agar surface [START_REF] Banaha | Spreading of liquid drops on agar gels[END_REF]. Flagella, whose deployment presumably depends on an appropriate fluid film on the agar surface, are essential for a later stage in the development of the bud and for driving dendrite migration (see below). Importantly, the entire process of bud formation and elongation of the radiating dendrites, up to lengths of 1.5 cm, occurs as a monolayer of cells. The cells in dendrites are distributed in an irregular mesh-like organization, including closely packed but clearly separated cells. Dendrites can be divided into two distinct regions, a long stem containing largely non-motile cells, which remarkably are maintained at an overall constant population density, and the extreme 1 mm at the tip where the population density increases sharply. This tip region contains hyper-motile cells that we term swarmers, which appear to constitute the 'motor' for elongation. Dead cells in dendrites or the MC are rare, perhaps less than 1%.

When dendrites reach approximately 1.5 cm in length, equivalent to 5-6 h following emergence of pre-dendrite buds, a dramatic switch from monolayered to multilayered dendrites slowly spreads progressively from the base of dendrites, as the swarm begins to develop the classical biofilm form ( at least 50 µm thick). These observations demonstrate that nutrients are in great excess for at least 24 h encompassing the swarming process and subsequent maturation of the bilayer. Moreover, we have shown that diluting the nutrients in the swarm plates at least 4-fold, prior to inoculation has little obvious effect on the pattern of swarming (unpublished data).

Recent studies [START_REF] Hamze | Identification of genes required for different stages of dendritic swarming in Bacillus subtilis, with a novel role for phrC[END_REF], using genetic analysis and fluorescent microscopy to measure the level of production (expression) from the gene encoding the major flagellum subunit in situ, have identified a specific subpopulation of hyper-flagellated cells ('swarmers' ). These are dominant in the formation of buds and then subsequently spearhead dendrites in the tips. These hyper-motile cells are in contrast to the cells forming the stem of dendrites that we term 'supporters'. We propose (manuscript in preparation) that supporters contribute to dendrite elongation by growth and division (multiplication), while swarmers actively drive extension from the tip, generating hydrodynamic forces dependent upon their hyper-motility. Thus, migration of the swarm front results from the co-operative action of two sub-populations to promote dendrite extension. In contrast to these mechanical forces, we note, however, that the piloting mechanism or guidance system that ensures radial migration, likely depends on a self-generated chemical gradient, but so far this remains completely unknown.

We shall now summarize a number of notable features of the migration process and consequent pattern development that should be taken into account when constructing mathematical models, if these are to describe the swarming process adequately. Migrating dendrites on encountering a large obstacle (like a cover slip or an E. coli colony) are induced to make a 90 o turn but then in most cases these return to the original radial direction. In addition, dentrites rarely merge and appear to avoid each other. This behaviour is consistent with the diffusion of a chemical repellent generated by the cells able to keep dendrites well separated. Significantly, the limited number of primary dendrites established at the initiation of swarming are usually supplemented at much later times by additional dendrites arising from the MC. This might indicate that the differentiation event to form swarmers (capable of breaking out of the MC) is based on a stochastic process that can occur repeatedly over time. In relation to the branching process, we have observed that branching can occur by tip splitting. However, a significant number of branches abort and remain very short and are often restricted to one side of the dendrite stem. All new branches tend quickly to adopt a radial direction during subsequent elongation. Importantly, while the overall frequency of branching increases towards the edge of the swarm plate, dendrite stems progressively become thinner (Fig. 8).

A surprising characteristic of the swarming process is the paradox that while cells are expected to grow and divide exponentially, as all cells do in a liquid culture, the rate of swarming migration remains constant. This, combined with the constant population density over most of the dendrite, clearly indicates that not all cells in the dendrites can be growing at the same rate. This would be an extremely unusual behaviour for a bacterial population and it is important to establish now which subpopulations may be subject to growth rate control.

Critical review of the correspondence between experiments and models

Whereas the models presented here clearly produce dendrites of various shapes, a critical analysis of the experimental data reveals that many features cannot be explained by these models. Moreover, as we will now discuss, these observations make it clear that an entirely new class of models will be needed for a detailed description of swarming in Bacillus subtilis and probably other bacteria. The major point concerns the mechanism of branch formation. The models (2.1), (2.2), and (2.3), as well as most of the models found in the literature, suppose that the proliferation of the bacteria is limited by the availability of a chemical nutrient which satisfies a diffusion equation. In some situations described in the literature this appears to be true, and convincing agreement between patterns from models and experiments has been obtained [START_REF] Golding | Studies of bacterial branching growth using reaction-diffusion models for colonial development[END_REF]. This however, may be an illusion. In all standard microbiological media, including Peptone -a so called poor medium employed in many early swarming studies -in which indeed the growth rate of cells is slow because the amino acids constituting the carbon source are not energy rich, many generations of growth can nevertheless be supported before nutrients become limiting. For the swarming experiments Figure 8: An experimental swarming pattern displayed by B. subtilis 168 on B-medium. Bacteria were inoculated in the centre of the plate and incubated for 24 h. Dendrites elongate radially from the central mother colony (approximately 3.5 mm per h) and begin branching after 1.5 cm. Highly reproducible patterns are obtained, characterized by increased frequency of progressively thinner branches. Dendrites generally appear to evade each other and rarely fuse. Side branches tend to be biased to one side and rather frequently abort after a relatively short distance. Interestingly, this motif, termed domain branching, is observed in mouse lung tissue [START_REF] Metzger | The branching programme of mouse lung development[END_REF]. Side branches frequently commence at 40 -90 o to the main branch, but then adopt a radial direction. described in our studies (see Section 3.), certainly, nutrients do not become limiting for growth, as shown by the continued visible increase in cell numbers for many hours after completion of the swarming process. Similarly, swarming in Proteus mirabilis is not controlled by nutrient limitation [START_REF] Esipov | Kinetic model of Proteus mirabilis swarm colony development[END_REF][START_REF] Rauprich | Periodic phenomena in Proteus mirabilis swarm colony development[END_REF].

The importance of this experimental observation stems from the fact that gradient in nutrient concentration is required in these models to create dendrites, since this promotes faster growth of bacteria in the tips that have easier access to the nutrients. In the absence of nutrient limitation, the question arises which physical or chemical effect pilots the outward migration of the bacteria and leads to branch formation. Two different potential alternative mechanisms can be envisaged for Bacillus subtilis.

A first hypothesis is suggested by the fact that mutants with reduced surfactin production swarm slowly or do not exhibit swarming. Moreover, the swarming process is accompanied by spatial gradients of surfactin concentration. Since surfactin is a surfactant, concentration gradients give rise to Marangoni forces, which have been shown to create branching patterns in the spreading of liquid droplets [START_REF] Troian | Fingering instabilities in thin wetting films[END_REF]. Moreover, many models to explain swarming assume that the surface of the agar is covered by a thin liquid film [START_REF] Golding | Studies of bacterial branching growth using reaction-diffusion models for colonial development[END_REF][START_REF] Bees | The interaction of thin-film flow, bacterial swarming and cell differentiation in colonies of Serratia liquefaciens[END_REF]. However, the reality is more complex since a simple experiment shows that a droplet of pure water deposited on the surface of the agar gel does not spread over the surface, but remains sessile [START_REF] Banaha | Spreading of liquid drops on agar gels[END_REF]. This obviously implies that the surface of the agar cannot be covered by a continuous liquid film. Nevertheless, in the presence of surfactin, a very thin film develops in the vicinity of the deposited droplet and expands slowly (the covered distance is approximately ∼ t 0.45 [START_REF] Banaha | Spreading of liquid drops on agar gels[END_REF]). While the presence of this film is necessary for swarming, presumably because this allows the movement of the bacteria, it is unlikely to provide a mechanism for the formation of dendrites. Indeed, the thickness of the film is close to that of prewetting films and these have never been reported to lead to the formation of dendrites on a solid liquid interface. Furthermore, the results of [START_REF] Banaha | Spreading of liquid drops on agar gels[END_REF] clearly indicate that the agar surface has a quite complex structure and that the wetting phenomena at this surface are not well understood. Therefore, a better understanding of these wetting properties is needed before incorporating them into new models, even at a qualitative level.

A second hypothesis, which was the basis for formulating models (2.4), is that the outward migration of the bacteria is driven by a long-range chemorepellent. Whereas, as yet, no substance generating such a chemotactic movement of the bacteria has been identified experimentally, the fact that dendrites avoid each other is consistent with the existence of a chemorepellent. As was shown in Section 2., simulations of such models can indeed produce dendritic structures. However, the mechanisms for branching and the overall growth of dendrites are quite different in the model and in the experiments.

In particular, the models presented here, as well as most of those in the literature, introduce two 'states' of the bacteria: 'active' cells which diffuse and reproduce, and 'passive' ones that do neither move nor reproduce. As was shown in Section 2., a characteristic feature of these models during branch formation are 'hot spots' of active bacteria that are located at the tips of the branches, whereas the main parts of the branches consist exclusively of inactive bacteria. As a consequence, the elongation of the branches takes place by cell division in the 'hot spots'. The experimental observations yield a quite different picture. Although the tips of the dendrites can be described as 'hot spots', characterized by a higher population density of bacteria, which in addition move much faster than the average population, the doubling time of the bacteria under the conditions considered here is much longer than the typical time for swarming, indicating that the driving mechanism for the swarming is cell migration and not cell division. Moreover, although not as highly motile, the bacteria in the 'stems' are by no means inactive. Some cells at least appear to perform a random-walk type motion with a global drift towards the tips that may support tip motion since cutting the stem with an obstacle stops the swarming at the head of the dendrite (unpublished observations, Orsay).

These observations have major non-trivial implications for the formulation of new models. This is related to the structure of the front of the swarm community. In the models presented here, the diffusion coefficient of the bacteria is always positive. In the 'hot spot', the balance between diffusion and cell division creates a propagating front solution that remains well localized. If the bacteria were to remain active, as experimentally observed, behind the tips, the diffusion process would lead to a spreading of the bacteria, and the gaps between the branches would disappear; only the transition to an inactive state, where bacteria neither move nor divide, maintains the stems intact. This conclusion remains valid even in models where the diffusivity of the bacteria depends on the local bacterial population density u and tends to zero when u tends to zero [START_REF] Müller | Morphological instability and dynamics of fronts in bacterial growth models with nonlinear diffusion[END_REF].

Consequently, a model must yield stable interfaces between the inside and the outside of the colony, even if all the bacteria are active. This necessarily requires some effect which leads to a 'cohesion' of the colony. In the Keller-Segel model, a short-range chemoattractant leads to the aggregation of the bacteria, but this model yields spot patterns, whereas stripes of active bacteria are unstable. This model therefore cannot describe stable branches. A class of models which can describe stable stationary and moving fronts is known from the physical description of phase transitions, the simplest example being the Cahn-Hilliard equation. In such models, two stable states (corresponding to the two thermodynamic phases in contact) are connected by a stable front solution of well-defined width. This front is maintained by the balance between a diffusion coefficient which becomes negative in the front and terms containing higher-order spatial derivatives to prevent the solution from developing singularities. While such models have numerous potentially useful properties for the description of bacterial community expansion, it is difficult at the present stage to find a well-justified equivalent to the free energy functional that underlies their mathematical structure.

In summary, many fundamental aspects of the swarming process are not well reproduced by the models available at present. Furthermore, as we have discussed above, some experimental observations indicate that the structure of the models has to be profoundly modified. Thus, dendrite elongation, involving exhaustion of nutrients by active cells at the swarm front, while attractive mathematically, appears untenable microbiologically. On the other hand, mathematical modelling has clearly predicted the participation of distinctive cell types involved in swarming, active leaders and passive cells forming the bulk of the dendrite. Experimental evidence from studies in P. mirabilis and now in B. subtilis (Hamze et al submitted), with the demonstration of distinctive cell types, swarmers and supporters, equivalent to active and passive types respectively, confirms the reality of these predictions. In addition, modelling studies more importantly make the exciting prediction that while active cells multiply, passive cells do not. Differential growth regulation of this type would be a novel concept in the field of bacterial communities. Nevertheless such regulation could explain some, so far, puzzling experimental observations. Future studies will be directed to testing this hypothesis, in addition to other experiments clearly needed to guide further model development. In particular, a much better understanding of the motion of a single bacterium and its interplay with the community is required. This necessitates further experiments focused on the analysis of the trajectories of single bacteria. In addition, a better understanding of the structure of the agar surface and its consequences for the local motility of the bacteria is desirable. Finally, models with several different population density fields of bacteria, reflecting distinct cell types, will certainly be required to account for the full complexity of the swarming process.

Figure 4 :

 4 Figure 4: Dynamics of the dendritic expansion as given by a numerical simulation of the Mimura model (2.3). Left: active cells modeled by the component u(x, t). Right: total population u + w.

Figure 5 :

 5 Figure 5: Dynamics of community expansion as given by a numerical simulation of a model (2.4), including cell population densities, chemoattractant and chemorepellent factors, but no nutrient limitation. Left: active cells modeled by the component u(x, t). Right: total population u + w.Compared to Figures2 and 4, the oriented drift creates more dynamical patterns where active spots can cross each other leading to an interwoven pattern.

Figure 6 :

 6 Figure 6: An example of the triangular mesh with 1024 elements. The meshes used for the numerical solutions are obtained by successive subdivisions with similar triangles and adjustment of the vertices to the curved boundary. Finer grids with up to 600.000 elements have been used.

Figure 7 :

 7 Figure 7: Example of numerical simulations of the Gray-Scott model (2.2) with the same set of parameters as in Fig. 1-3 , except we have used the ratio d v = 20d u (this makes the pattern denser compared to the ratio 100 used previously), using different triangular meshes; from left to right: 65536 elements, 262144 elements, 589824 elements.

Acknowledgements

The authors thank Agence Nationale pour la Recherche (grant ANR-05-blanc 0138) for financial support. SJS and IBH also wish to acknowledge the support of Université Paris-Sud.


Yue Xu

Chentao Yan

Chunlin Du

Kai Xu

Yixuan Li

Miaojun Xu
email: xumiaojun@nefu.edu.cn

Serge Bourbigot

Gaelle Fontaine

Bin Li

Lubin Liu
email: liulubin@nefu.edu.cn

High-strength, thermal-insulating, fire-safe bio-based

Keywords: Bio-based tubular aerogel, 3D network structure, Thermal insulation, Fire safety, High strength

come

Introduction

In modern society, the demand (≈23 billion square meters in 2022 alone) for building insulation, energy storage and sound absorption materials is growing exponentially every year [START_REF] Hauenstein | Bio-based polycarbonate as synthetic toolbox[END_REF][START_REF] Rathore | Thermal performance of the building envelope integrated with phase change material for thermal energy storage: an updated review[END_REF][START_REF] Wu | Broadband multispectral compatible absorbers for radar, infrared and visible stealth application[END_REF]. Organic insulation materials are widely used in the fields of insulation and energy storage due to their advantages of light weight, low thermal conductivity and easy molding [START_REF] Apostolopoulou-Kalkavoura | Thermally insulating nanocellulose-based materials[END_REF][START_REF] Yang | Poly(vinylalcohol)/chitosan-based highstrength, fire-retardant and smoke-suppressant composite aerogels incorporating aluminum species via freeze drying[END_REF][START_REF] Gu | A lightweight, elastic, and thermally insulating stealth foam with high infrared-radar compatibility[END_REF]. Nevertheless, the traditional organic insulation materials are from petrochemical-based materials such as extruded polystyrene board, foamed polystyrene board, polyurethane foam, and their high energy consumption and non-degradation in the production and processing process bringing huge energy and environmental pressure to society [START_REF] Liu | 2D MXenes for fire retardancy and fire-warning applications: promises and prospects[END_REF]. Meanwhile, the traditional organic insulation materials are often flammable and release a large number of toxic gases during combustion process, causing a lot of economic losses (direct loss of 5.85 billion in 2022 of China) and casualties to human beings [START_REF] Yan | Experimental study on the width and pressure effect on the horizontal flame spread of insulation material[END_REF]. Therefore, the development of biodegradable, fire-safe, recyclable bio-based organic insulation materials in line with current sustainable development strategies has become a top priority today.

With the proposal of the dual carbon strategy, a potential and sustainable solution is offered through environmentally friendly and abundant natural products as J o u r n a l P r e -p r o o f substitutes to petroleum-based insulation materials. Hence, bio-based raw materials such as cellulose [START_REF] Wang | High-resilience cotton base yarn for antiwrinkle and durable heat-insulation fabric[END_REF], lignin [START_REF] Kuranska | Synthesis of thermal insulating polyurethane foams from lignin and rapeseed based polyols: A comparative study[END_REF], alginate [START_REF] Wang | Temperature-triggered fire warning PEG@wood powder/carbon nanotube/calcium alginate composite aerogel and the application for firefighting clothing[END_REF], and chitosan [START_REF] Chen | Morphology control of eco-friendly chitosan-derived carbon aerogels for efficient microwave absorption at thin thickness and thermal stealth[END_REF] were used to prepared sustainable organic insulation materials, which showed great promise in replacing petrochemical-based insulation materials with similar properties. Among these biomaterials, cellulose, as a bio-based raw materials with the largest reserves on earth, is the most promising alternative material, and it has been widely used in insulation, adsorption, energy storage, electronic and electrical fields [START_REF] Lou | Fully biobased surfacefunctionalized microcrystalline cellulose via green self-assembly toward fire-retardant, strong, and tough epoxy biocomposites[END_REF][START_REF] Chen | Nanocellulose: a promising nanomaterial for advanced electrochemical energy storage[END_REF][START_REF] Chen | Recent Progress on Nanocellulose Aerogels: Preparation, Modification, Composite Fabrication, Applications[END_REF][START_REF] Gu | Lignocellulose-derived hydrogel/aerogel-based flexible quasi-solid-state supercapacitors with high-performance: a review[END_REF]. Nevertheless, with the continuous development of high-end manufacturing, cellulose-based materials still face numerous challenges in structural construction and multifunctionalization. Usually, the structure of materials decides their performance, and the natural tubular structure affects its insulation properties in nature, such as the feathers of birds and the hair of polar bears, etc. Previous studies on aerogel materials have shown that the construction of tubular structures was beneficial to the thermal insulation of materials [START_REF] Hu | Hollow-structured materials for thermal insulation[END_REF][START_REF] Wicklein | Thermally insulating and fire-retardant lightweight anisotropic foams based on nanocellulose and graphene oxide[END_REF]. However, the traditional preparation methods for cellulose-based tubular aerogels, such as the template method and the introduction of inorganic tubular materials, are more complicated and have a greater impact on the comprehensive performance of materials [START_REF] Dong | Biomass poplar catkin fiber-based superhydrophobic aerogel with tubular-lamellar interweaved neurons-like structure[END_REF]. Therefore, it is a new way to search a natural, recyclable and sustainable tubular bio-based raw material as a substrate for cellulose-based aerogels. Kapok fibers (KF), as a natural hollow tubular structure with light weight, renewable, degradable and superior biocompatibility, is commonly used in textile, adsorption, solar steam generation and sound insulation fields [START_REF] Thilagavathi | Oil sorption and retention capacities of thermallybonded hybrid nonwovens prepared from cotton, kapok, milkweed and polypropylene fibers[END_REF][START_REF] Wang | Coated kapok fiber for removal of spilled oil[END_REF][START_REF] Mu | Conductive hollow kapok fiber-PPy monolithic aerogels with excellent mechanical robustness for efficient solar steam generation[END_REF][START_REF] Xiang | Investigation on sound absorption properties of kapok fibers[END_REF]. Meanwhile, KF possessed a low thermal conductivity due to its own tubular structure and can be designed for the construction of tubular bio-based insulation aerogel after surface treatment.

Although natural tubular bio-based aerogels have been constructed using the hollow structure of kapok, the poor fire safety and compression strength of KF aerogels still constrained their application in insulation and energy storage fields due to their own structural composition (64% cellulose and 13% lignin) in practical applications. In the past few years, scholars have conducted a lot of research on flame retardant cellulose-based aerogels [START_REF] Ahankari | Recent developments in nanocellulose-based aerogels in thermal applications: A review[END_REF]. The introduction of inorganic fillers or intumescent flame retardants are the most commonly used modification methods [START_REF] Zhang | Phosphorus containing group and lignin toward intrinsically flame retardant cellulose nanofibril-based film with enhanced mechanical properties[END_REF]. Among them, the inorganic fillers for flame retardant cellulose-based aerogels mainly included metal hydroxide [START_REF] Han | Flame Retardant, Heat insulating cellulose aerogels from J o u r n a l P r e -p r o o f waste cotton fabrics by in situ formation of magnesium hydroxide nanoparticles in cellulose gel nanostructures[END_REF], montmorillonite [START_REF] Koklukaya | Superior Flame-resistant cellulose nanofibril aerogels modified with hybrid layer-by-layer coatings[END_REF], hydroxyapatite [START_REF] Yang | Fabrication of fully bio-based aerogels via microcrystalline cellulose and hydroxyapatite nanorods with highly effective flameretardant properties[END_REF] and nano clay [START_REF] Gupta | Flame retardant and thermally insulating clay based aerogel facilitated by cellulose nanofibers[END_REF]. However, the incorporation of inorganic fillers usually decreased the strength and thermal J o u r n a l P r e -p r o o f insulation properties of aerogel composites and increased their density of the materials.

Compared with inorganic fillers, intumescent flame retardants exhibited higher flame retardant efficiency in cellulose-based aerogels [START_REF] Zhao | Strong, Thermally superinsulating biopolymer-silica aerogel hybrids by cogelation of silicic acid with pectin[END_REF]. But the addition of intumescent flame retardants still cannot give aerogel composites high strength, low density and thermal conductivity while obtaining excellent fire safety properties. Therefore, it is urgent to develop a bio-based aerogel composite with excellent fire safety, light weight, thermal insulation, high strength and biodegradability to meet the needs of practical applications.

Herein, the 3D network construction of KF tubular structure by biodegradable binder PVA and biguanide phosphonate (BGP) significantly improved the thermal insulation, mechanical and fire safety properties of materials while ensuring the light weight and degradability of aerogel composites (Fig. 1). Due to the existence of waxy layer on the surface of KF, it was firstly micro-treated and processed into 150~200 μm hollow tubular fibers by using unique cutting method to ensure the full component utilization of KF. PVA was used as a dispersing binder to ensure the high filling amount of KF, which made the aerogel construct a tubular-based 3D network structure.

Moreover, with the introduction of flame retardant cross-linker BGP, the KF-PVA-BGP suspension exhibited a high viscosity at room temperature and thinning in shear (Fig. S1), which ensured that KF-PVA-BGP aerogel can be readily prepared by freezemolding and -drying at any time. KF-PVA-BGP aerogel presents low shrinkage and high compressive strength due to their intermolecular interaction. This method uses recyclable and renewable bio-based KF as the matrix to prepare a novel biodegradable aerogel that integrates light weight, high strength, thermal insulation and fire safety, and the aerogel composite can be used as plant fertilizer during degradation process, which is in sharp contrast to non-degradable and unsustainable petrochemical-based plastics.

J o u r n a l P r e -p r o o f

Experimental Section

Preparation of KF and flame retardant aerogels

First, the surface of KF was simply dewaxed with 8 g/L sodium hydroxide solution [START_REF] Zhou | Lotus leaf-inspired and multifunctional Janus carbon felt@ Ag composites enabled by in situ asymmetric modification for electromagnetic protection and low-voltage joule heating[END_REF], and washed to pH=7. As shown in Fig. 1, the rough tube structure on KF surface was exposed. Then, the PVA is sequentially used in different proportions (1%, 2%, 3%, 4%, 5%, 6%, 7%) to rebuild the cross-linking network structure of KF. After freezedrying at -60 o C for 72 h, KF aerogels were obtained. The prepared KF-PVA aerogels were named as KF, KF-PVA1, KF-PVA2, KF-PVA3, KF-PVA4, KF-PVA5, KF-PVA6, KF-PVA7, respectively. Meanwhile, biguanide phosphonate (BGP) solutions with different concentrations were prepared according to the proportions of 2%, 4%, 6%, 8% and 10%, respectively, and then fully mixed with 5% KF-PVA solution to form J o u r n a l P r e -p r o o f hydrogels. After freeze-drying at -60℃ for 72 h, flame retardant KF aerogels were obtained, which were named as KF-PVA5-BGP2, KF-PVA5-BGP4, KF-PVA5-BGP6, KF-PVA5-BGP8, KF-PVA5-BGP10, respectively.

Characterization

Scanning electron microscopy (JSM-7500F SEM, Japan) was used to collect the microstructures of the aerogels and char residue, and the EDS mapping was used to analyze element dispersion. Fourier transform infrared spectroscopy (FTIR) was recorded from 400 to 4000 cm -1 with a resolution of 4 cm -1 on Nicolet Avator 360 spectrometer (Madison, America). Solid-state 31 P NMR was analyzed on the AVANCE III HD-500 MHz spectrometer (Bruker, Germany). The C, N, O and P elements of aerogels and char residues were obtained from X-ray photoelectron spectrometer (XPS) (Thermofisher, America), and the thickness of XPS detection was about 10 nm. The porosity of KF aerogel was calculated from the apparent density (ρ) of the material according to equation 1. The a, b and c are percentages of KF, PVA and BGP, respectively.

𝑝𝑜𝑟𝑜𝑠𝑖𝑡𝑦 = (1 - ρ a×ρKF+b×ρPVA+c×ρBGP ) × 100% (1)
The mechanical performance was measured by universal testing machine J o u r n a l P r e -p r o o f

Results and discussion

Construction and microstructure of KF aerogels

In this research, multifunctional KF aerogels were successfully prepared by the method of mixing dispersion-freezing casting-lyophilized molding. First of all, in order to ensure the full utilization of KF, we only carried out a simple microtreatment to remove the wax on its surface. In Fig. S1, the waxy peaks of KF at 1741 and 1249 cm - 1 disappeared obviously after alkali treatment and the aromatic groups in the structure were retained, which was beneficial to the application of KF with full composition and high value. The dewaxed KF can be uniformly dispersed with PVA and BGP in deionized water through mechanical stirring and the natural hollow KF was strongly bonded together due to the hydrogen bonding action between PVA, BGP and KF.

Therefore, KF aerogels were endowed with excellent thermal insulation, flame retardancy and mechanical properties.

Fig. 2 showed the morphology of KF after treatment. KF was an obvious hollow structure with a diameter of approximately 15 μm (Fig. 2b). After alkali treatment, the surface of KF became rough due to the removal of wax, ash and water-soluble substances on the surface. In addition, the surface of treated KF possessed polyhydroxy structures and it can lap itself into an aerogel through freeze-forming and freeze-drying (Fig. 2b). However, the formed aerogel was easily broken and slag-dropped, and its volume shrinkage rate was relatively large, which cannot meet the needs of daily applications. With the introduction of binder PVA, the volume shrinkage of the KF-PVA aerogels were significantly decreased (Fig. S2). When the addition of PVA was 5 wt%, the volume retention rate of KF-PVA aerogel remained at approximately 91% and it basically did not change with the improvement of PVA content. The hollow structure of KF was obviously preserved with the introduction of PVA and the adhesion of treated KF was obviously improved (Fig. 2c). KF-PVA5 aerogels exhibited the lowest density (0.0587 g•cm -3 ) due to its improved volume retention rate of KF-PVA aerogels and the 3D network structure between PVA and KF. After introducing BGP into KF-PVA5 aerogels, the KF-PVA5-BGP aerogels still maintained a low volume shrinkage (Fig. S2)

and the porosity of KF-PVA5-BGP aerogels was subsequently improved with the introduction of BGP (Fig. 2g). Meanwhile, because of the hydrogen bonding between BGP, PVA and KF, the adhesion between KF and PVA was stronger and the 3D network significantly higher than that of BGP (Fig. 2h). In Fig. S3, the P2p peaks combined with the above analysis further indicated the successful introduction of BGP and the role of intermolecular hydrogen bonding.

Mechanical performance and enhancement mechanism analysis

The KF aerogels can be tailored and processed into various complex shapes according to our requirements (Fig. 3d). Besides, the KF aerogels possessed a low density and can be stand on thin petals without deformation (Fig. 3c). Compared with traditional aerogels, KF-PVA-BGP aerogels not only ensured full-component and highload utilization of biomass fiber resources, but also have high strength and low density [START_REF] Apostolopoulou-Kalkavoura | Thermally insulating nanocellulose-based materials[END_REF][START_REF] Yeo | Structurally controlled cellular architectures for highperformance ultra-lightweight materials[END_REF]. The volume retention rate and mechanical resistance of KF-PVA aerogels were significantly enhanced compared with the friability and shrinkage of pure KF aerogel (Fig. S2 and3d). When the introduction of PVA was 5 wt%, the mechanical strength of KF-PVA5 aerogel exhibited the highest value (1.29 MPa). Meanwhile, the structure of KF-PVA aerogel was reinforced with the introduction of BGP and the compressive strength of KF-PVA5-BGP10 was reached up to 1.64 MPa. 1 g specimen could withstand a mass 1500 times larger than itself without any deformation, which was relatively high in the reported organic aerogels (Fig. 3c) [START_REF] Aoudi | Recent advances and future perspective on nanocellulose-based materials in diverse water treatment applications[END_REF]. The improvement in strength for KF-PVA-BGP was attributed to the effect of their intermolecular hydrogen bonds and the corresponding enhancement mechanism was illustrated in Fig. 3e. PVA and BGP served as oxygen-and nitrogen-rich structures respectively and the introduction of N and O could generate hydrogen bonds with -OH groups on the surface of KF, leading to form the 3D network structure in aerogel and improving effectively their compression strength [START_REF] Liu | Bioinspired J o u r n a l P r e -p r o o f strong, tough, and biodegradable poly (vinyl alcohol) and its applications as substrates for humidity sensors[END_REF][START_REF] Liu | Dynamic nanoconfinement enabled highly stretchable and supratough polymeric materials with desirable healability and biocompatibility[END_REF]. Secondly, although KF-PVA-BGP aerogels maintained a relatively low density, their density was slightly increased compared to KF-PVA aerogels, which was also positive for the enhancement of mechanical properties of KF-PVA-BGP aerogels.

J o u r n a l P r e -p r o o f

Thermal insulation analysis

Thermal insulation performance is an important indicator for the practical application of aerogel in the field of energy saving and thermal insulation. To study the J o u r n a l P r e -p r o o f thermal insulation of high-strength aerogels constructed from natural hollow tubular fibers, the thermal conductivity and thermal images of KF aerogels at different temperatures were evaluated. The thermal conductivity of KF-PVA aerogels decreased gently due to the formation of 3D network structure in aerogels with the increase of PVA addition (Fig. 4a-b). The introduction of BGP made the hydrogen bond between

Fire Hazard Analysis

The fire safety of organic aerogels is crucial in the field of energy saving and thermal insulation. Therefore, the flame retardancy of aerogels were evaluated by direct ignition method (ignite with a lighter for 1 s). Pure KF aerogel burned rapidly once it was near an ignition source and started to smolder after the open flame ended until it burned to a white flocculent ash (Fig. 5a). When the flammable binder PVA was introduced into the KF aerogels, the flammability of the KF-PVA5 aerogels significantly increased and its burning time improved accordingly. In contrast, the KF-PVA5-BGP10 aerogels achieved a quick self-extinguishing effect with the introduction of BGP and J o u r n a l P r e -p r o o f retained its original morphology after three ignitions. This possessed important implications for expanding the application value of high-strength organic KF aerogels.

In order to further analyze the effect of PVA and BGP on the combustion behaviour of KF aerogels, the cone calorimetry (CC) tests of KF, KF-PVA5 and KF-PVA5-BGP aerogels were performed and the related test data were shown in Fig. 5 and Table S1.

Once pure KF aerogels was ignited, it showed a high PHRR (peak of heat release rate) value of 213.93 kW•m -2 and its THR (total heat release) was up to 19.21 MJ•m -2 . The PHRR and THR of KF-PVA5 aerogels were enhanced due to the introduction of flammable PVA. Interestingly, the introduction of flame retardant reinforcement BGP significantly reduced the PHRR and THR of KF-PVA5 aerogels. When the introduction of BGP was only 10 wt%, the PHRR and THR of KF-PVA5-BGP10 aerogels were decreased by 56.0% and 30.5%, respectively. This fact was mainly attributed that BGP promoted the cross-linking and charring of polyhydroxy KF-PVA matrix, which exerted the shielding effect in condensed phase. It was also evidenced by the early degradation and increased degradation rate of the aerogel composites in TG analysis of Fig. 5h-i.

In actual fires, the generation of smoke and toxic gases can easily lead to asphyxiation and they are more lethal than heat release. The PSPR (peak of smoke production rate) and TSP (total smoke production) of KF-PVA5-BGP10 aerogels were decreased by 74.3% and 86.1% compared with those of KF aerogels (Fig. 5f-g). This significant decrease was ascribed to the excellent catalytic charring properties of BGP and the shielding effect of constituent char layer. It was also illustrated through the high residual mass in the TG analysis of Fig. 5h. In general, a high residual mass meant that less fuel was used for combustion. Thus, the KF-PVA-BGP aerogels showed excellent fire safety compared with traditional thermal insulation materials.

J o u r n a l P r e -p r o o f To make the fire hazard of aerogels clearer, their FPI (fire performance index), FGI (fire growth index) and FRI (flame retardancy index) was calculated according to J o u r n a l P r e -p r o o f equations (2-4). Theoretically, flame retardant materials with a high FPI and low FGI demonstrated good fire safety. As shown in Table S1, the FPI value of KF-PVA-BGP aerogels increased and the FGI values decreased subsequently with the introduction of BGP addition, indicating that the fire hazard of KF-PVA-BGP aerogels was significantly reduced. Meanwhile, the FRI was used to evaluate the flame retardant performance of materials. The FRI of KF-PVA-BGP aerogels gradually increased with the introduction of BGP (Table S1). It meant that BGP efficaciously improved the flame retardant performance of aerogel composites. In summary, compared to conventional petrochemical-based energy saving and thermal insulation materials (e.g. polyurethane, polystyrene, etc.) [START_REF] Muhammed | Emerging trends in flame retardancy of rigid polyurethane foam and its composites: A review[END_REF], KF-PVA-BGP aerogels exhibited excellent fire safety performance in addition to their biodegradability. This novel aerogel materials possessed important research implications for reducing the risk to people and extending their escape and rescue time during a fire.

FPI = TTI PHRR (2) FGI = PHRR TPHRR (3) FRI = [THR× ( PHRR TTI )]
Neat aerogel

[THR× ( PHRR TTI )]

Aerogel composites (4)

Flame retardant mechanism analysis

The above fire hazard analysis indicated that KF-PVA-BGP aerogels exhibited excellent fire safety performance. In the following section, the flame retardant mechanism of KF-PVA-BGP aerogels was revealed through the combined gas-and condensed-phase analysis.

The photos, microscopic morphology and chemical structure composition of char residues for KF, KF-PVA5 and KF-PVA-BGP aerogels after CC tests were shown in Fig. S4, 5 and 6. Pure KF and KF-PVA5 only remained white flocculent residue under strong radiation. In Fig. 6a, FTIR analysis indicated that they were oxalic carbonates (υCO3:1448~1000 cm -1 , δCO3: 960~660 cm -1 ) [START_REF] Wang | Lignocellulosic biomass pyrolysis mechanism: A stateof-the-art review[END_REF]. With the introduction of flame retardant enhancer BGP, the white grassy ash substance resembling KF aerogels disappeared obviously and the residual mass of KF-PVA-BGP aerogels was J o u r n a l P r e -p r o o f significantly improved. When the introduction of BGP was 10 wt%, KF-PVA5-BGP10 aerogels formed a complete and coherent char layer after CC tests. The residual char of KF-PVA5-BGP10 aerogels had a compact and relatively dense morphological structure and retained the original hollow structure of KF after CC tests, which effectively acted as a heat shield and prevented the exchange of combustibles in condensed phase (Fig. 5j-l). The FTIR analysis of char residues for KF-PVA5-BGP10 exhibited several absorption peaks at 1592 cm -1 (aromatic ring), 1186 cm -1 (P=O) and 960 cm -1 (P-O), demonstrating the formation of a phosphorus-containing polyaryl char layer [START_REF] Liu | A reactive copper-organophosphate-MXene heterostructure enabled antibacterial, selfextinguishing and mechanically robust polymer nanocomposites[END_REF][START_REF] Liu | Facile synthesis of an efficient phosphonamide flame retardant for simultaneous enhancement of fire safety and crystallization rate of poly (lactic acid)[END_REF][START_REF] Jia | Effects of BN/GO on the recyclable, healable and thermal conductivity properties of ENR/PLA thermoplastic vulcanizates[END_REF].

Meanwhile, EDS mapping and XPS analysis showed that the phosphorus-containing (13.14 wt%) compounds were homogeneously dispersed in condensed phase after CC tests. The BE (binding energy) peaks of O1s and P2p spectra for KF-PVA5-BGP10

proved the existence of O=P-O and PO3 (Fig. 6 d,e). Combined with the BE peaks of C1s spectra, BGP produced pyro/polyphosphate during decomposition, which effectively promoted the generation of cross-linked char layers. This allowed KF to maintain its original tubular morphology and thus exerted an excellent shielding role in condensed phase. and 3567 cm -1 ), hydrocarbons (2926 and 2898 cm -1 ), CO2 (2362 and 2322 cm -1 ), CO (2182 and 2111 cm -1 ), carbonyls (1744 cm -1 ), aromatics (1508 cm -1 ), ethers (1271 and 1105 cm -1 ) [START_REF] Liu | Economical and facile synthesis of a highly efficient flame retardant for simultaneous improvement of fire retardancy, smoke suppression and moisture resistance of epoxy resins[END_REF][START_REF] Xu | Bioinspired, strong and tough nanostructured poly(vinyl alcohol)/inositol composites: How hydrogen-bond cross-linking works?[END_REF][START_REF] Huang | A dual-band transceiver with excellent heat insulation property for microwave absorption and low infrared emissivity compatibility[END_REF][START_REF] Liu | Multifunctional polyurethane sponge coatings with excellent flame retardant, antibacterial, compressible, and recyclable properties[END_REF]. When the polyhydroxyl binder PVA was introduced, the pyrolysis products of KF-PVA aerogels had no obvious change except for the increased absorption peak of water. Nevertheless, as shown in Fig. S5, the amount of pyrolysis products (especially hydrocarbons (2898 cm -1 ), carbonyls (1744 cm -1 ) and esters (1105 cm -1 )) of KF-PVA-BGP was significantly lower than those of KF and KF-PVA aerogels with the introduction of BGP, which indicated that KF-PVA5-BGP10 aerogels generated less flammable gas during thermal degradation, resulting a lower fuel feeding and decreasing the combustion process of aerogel composites. Meanwhile, the introduction of BGP promoted the dehydration and charring of KF-PVA aerogel in advance and thus KF-PVA5-BGP10 aerogels produced a lot of water during their decomposition compared with KF, and the water and ammonia gas generated from the decomposition of BGP exerted the dilution effect in vapor phase. In addition, the catalytic carbonization of BGP resulted in relatively high content of aromatic compounds at the beginning of degradation, which was conducive to the formation of a hard and cross-linked highquality char layers of KF-PVA-BGP aerogels. The shielding effect of char layer and the increase of residual mass significantly reduced the emission of toxic gas CO and effectively enhanced the smoke and toxicity suppression performance of organic aerogels.

J o u r n a l P r e -p r o o f Based on the above analysis, a possible flame retardant mechanism of KF-PVA-BGP was illustrated in Fig. 7h. Once the KF-PVA-BGP aerogels were burned, the decomposition of BGP produced phosphoric acid compounds, which promoted the dehydration and charring of the KF-PVA matrix. Ammonia and other non-combustible gases from the combustion of aerogel composites effectively exerted dilution role in gas phase, decreasing the burning intensity of aerogels. Meanwhile, KF-PVA-BGP aerogels formed a compact, homogeneous and dense tubular char layer, and its shielding effect with the aerogel itself prevented the exchange of heat, combustible volatiles and oxygen in condensed phase. Thus, KF-PVA-BGP aerogels presented excellent fire safety properties.

Multifunctional thermal insulation aerogels applied in cold conditions

In the application process, the actual insulation performance of energy conservation and heat shielding materials is very important. Currently, the commonly J o u r n a l P r e -p r o o f used wall insulation materials in the markets are mainly petrochemical-based polystyrene (PS) boards. We prepared a simple house model by simulating the hot water heating conditions in the northern cold environment, and compared the thermal insulation performance of biodegradable KF-PVA5-BGP10 aerogels and commercial PS boards (Fig. 8c). KF-PVA5-BGP10 aerogels and PS foam possessed similar thermal insulation performance and the hot water could only drop to 30 o C in 5 h compared with pure wooden buildings (Fig. 8e). Nevertheless, although the PS foam acquired in the market was treated with flame retardant, it was very easy to be ignited under strong ignition source and its combustion process generated a large amount of smoke, which caused asphyxiation and death of human beings during fire. Compared with commercial PS foam, KF-PVA5-BGP10 bio-based aerogels not only has similar thermal insulation performance, but also exhibited excellent flame retardancy. KF-PVA5-BGP10 was rapidly self-extinguishing under twice ignition of strong fire source, which was essential to enhance the fire safety performance of building insulation composites.

Meanwhile, KF, as a light filling material, possessed good mildew resistance and it played a positive role on its use as a wall insulation material. Therefore, KF-PVA5-BGP10 aerogels was successfully endowed with excellent thermal shielding, mechanical, flame retardant and smoke-suppression properties compared with traditional petrochemical-based PS boards by using its own hollow structure and 3D network construction while maintaining its biodegradability (Fig. 8h,i) [START_REF] Yang | Poly(vinylalcohol)/chitosan-based highstrength, fire-retardant and smoke-suppressant composite aerogels incorporating aluminum species via freeze drying[END_REF]. This was a crucial support for the implementation of the global dual carbon strategy.

J o u r n a l P r e -p r o o f

Conclusions

In conclusion, we prepared a lightweight, insulating, robust, recyclable,

Fig. 1 .

 1 Fig. 1. Construction process and multifunctionality of KF aerogel composites.

2. 1

 1 . Materials KF were supplied by Husheng Trading Co., Ltd. (Nanjing, China). NaOH was purchased from Fuchen Chemical Reagent Co., Ltd. (Shanghai, China). Both polyvinyl alcohol (PVA-1788) and biguanide phosphonate (BGP) were obtained from Aladdin Reagent Co., Ltd. (Shanghai, China).

(

  Shenzhen Reger Instrument Company, China). The thermal conductivity of aerogels was characterized by TPS 2500s Hot Disk Thermal Constant Analyzer (Sweden) at 25 o C with a heating power of 20 mW and time of 40 s. Each specimen was tested at least 5 times, and the average value was used as the result. The thermographic images were obtained by a thermal imager of Testo 869 (Testo, Germany). Cone calorimetry (CC) tests were performed on a cone calorimeter tester (West Sussex, UK) under the heat flux of 35 kW•m -2 . The sample dimensions were 100 mm × 100 mm × 4 mm. Thermogravimetric analysis (TGA) was carried out on a STA 6000-SQ8 thermal analyzer with heating rate of 10 o C•min -1 under nitrogen atmosphere. TG-IR of the samples was measured on the Perkin Elmer STA 4000-SP3 analyzer under the N2 atmosphere and 10 mg of aerogel composites were heated to 800 o C at 10 o C•min -1 .

  J o u r n a l P r e -p r o o f structure was more obvious. It was beneficial for KF-PVA5-BGP aerogels to obtain excellent mechanical and thermal insulation properties.

Fig. 2 .

 2 Fig. 2. SEM images of (a) untreated KF, (b) KF aerogel, (c) KF-PVA 5 and (d) KF-PVA 5 -BGP 10 aerogels; (e) FTIR spectra of the KF, KF-PVA 5 and KF-PVA 5 -BGP 10 aerogels; (f) 31 P solid-state NMR spectra of BGP and KF-PVA 5 -BGP 10 aerogel; (g) Porosity of KF and KF-PVA-BGP aerogels; (h) High-resolution C1s XPS spectra of BGP and KF-PVA 5 -BGP 10 aerogels; (i) Highresolution N1s XPS spectra of KF and KF-PVA 5 -BGP 10 aerogel.The FTIR, 31 P solid-state NMR and XPS spectra of KF, BGP, KF-PVA5 and KF-PVA5-BGP aerogels were shown in Fig.2. The absorption peak of KF aerogel at 3390 cm -1 was attributed to the stretching vibration of -OH (Fig.2e). The characteristic peak of aromatic structure at 1598 cm -1 indicated that lignin in KF was preserved after NaOH treatment. The absorption peaks at 1162, 1112 and 1056 cm -1 were ascribed to the stretching vibration of C-O in KF[START_REF] Zhou | Lotus leaf-inspired and multifunctional Janus carbon felt@ Ag composites enabled by in situ asymmetric modification for electromagnetic protection and low-voltage joule heating[END_REF][START_REF] Zong | Preparation of TiO2/cellulose nanocomposites as antibacterial bio-adsorbents for effective phosphate removal from aqueous medium[END_REF][START_REF] Khan | A durable, flexible, large-area, flame-retardant, early fire warning sensor with built-in patterned electrodes[END_REF]. With the introduction of PVA and BGP, the stretching vibration peak of -OH gradually shifted to the lower band, which indicated the generation of hydrogen bonds between KF and PVA and BGP. Compared with KF and KF-PVA5 aerogels, the new characteristic peak of C=N appeared at 1665 cm -1 in KF-PVA5-BGP10 and it demonstrated the successful doping of BGP. Meanwhile, the 31 P solid-state NMR of BGP indicated that its P chemical shift was 4.08 ppm and there was a small spike at 0.38 ppm due to the existence of a small amount of monophosphate in

Fig. 3 .

 3 Fig. 3. Stress-strain curves of (a) KF-PVA and (b) flame retardant KF-PVA-BGP aerogels; (c) Images of KF-PVA 5 -BGP 10 sample under the load of 1.5 kg without deformation; (d) Compressive strength of different KF aerogels and the various complex shapes of KF-PVA-BGP aerogels; (e) Schematic diagram of the mechanical enhancement mechanism of KF-PVA-BGP aerogels.

Fig. 4 .

 4 Fig. 4. Thermal conductivity diagram of (a) KF and (b) KF-PVA-BGP aerogel samples; Side view pseudo-color thermal image of (c) KF and (d) KF-PVA 5 -BGP 10 aerogel with a height of 1.6 cm after heating at 200 o C for 10min; Top side pseudo-color thermal images of the KF-PVA 5 -BGP 10 with a height of 1.6 cm after heating for 1, 5 and 10 min on a hot plate held at (e) 50 o C, (f) 100 o C, (g) 150 o C and (h) 200 o C; (i) Temperature difference (ΔT) curves between the surface of hot stage and KF-PVA 5 -BGP 10 .

Fig. 5 .

 5 Fig. 5. Video screenshot of the combustion process of (a) KF, (b) KF-PVA 5 and (c) KF-PVA 5 -BGP 10 aerogels after ignition; (d) HRR, (e) THR, (f) SPR and (g) TSP curves of KF, KF-PVA 5 , KF-PVA 5 -BGP 6 and KF-PVA 5 -BGP 10 aerogels after CC tests; (h) TG and (i) DTG curves of KF, KF-PVA 5 and KF-PVA 5 -BGP 10 aerogels under N 2 ; (j-l) SEM images of residual char after CC tests for KF-PVA 5 -BGP 10 aerogels; (m) SEM image and (n-r) EDS Mapping of char residues for KF-PVA 5 -BGP 10 aerogels.

Fig. 6 .Fig. 7 .

 67 Fig. 6. (a) FTIR spectra of char residues for KF, KF-PVA 5 and KF-PVA 5 -BGP 10 aerogels after CC tests; XPS (b) survey, (c) C1s, (d) O1s and (e) P2p spectra of residual char of KF-PVA 5 -BGP 10 aerogel.

Fig. 7 .

 7 Fig. 7. 3D TG-FTIR spectra of pyrolysis products for (a) KF, (b) KF-PVA 5 and (c) KF-PVA 5 -BGP 10 aerogels; Time-dependent FTIR spectra of pyrolysis products for (d) KF, (e) KF-PVA 5 and (f) KF-PVA 5 -BGP 10 aerogels; (g) FTIR spectra of pyrolysis products for KF, KF-PVA 5 and KF-PVA 5 -BGP 10 aerogels at 340 o C; (h) Schematic diagram of the flame-retardant mechanism of KF-PVA-BGP aerogels.

Fig. 8 .

 8 Fig. 8. Simulated small room built with (a) KF-PVA-BGP, (b) PS board, and (c) wood board, respectively; (d) Hydrothermal simulation of heating environment in northern cold region; (e) Curves of temperature change over time under KF-PVA-BGP insulation boards, PS boards, wood boards and air conditions; Comparison of burning video screenshot of model house built by commercial (f) PS boards and (g) KF-PVA-BGP insulation boards after ignition; (h) Illustrations of KF-PVA-BGP boards used for roofing, external wall panels and floor insulation materials of the energy-saving buildings; (i) Comparison of the comprehensive performance of KF-PVA-BGP aerogels and commercial PS boards by a radar chart.

  biodegradable and excellent fire safety tubular aerogel with a 3D network structure by the now mature freeze-drying technique. KF-PVA-BGP suspensions possessed a high J o u r n a l P r e -p r o o f apparent viscosity and KF-PVA-BGP aerogels can be manufactured easily, enabling high filling and high value utilization of bio-based KF. Compared with commercial PS insulation boards, the KF-PVA-BGP composites exhibited higher compression strength and excellent thermal insulation properties because of the construction of 3D network for tubular aerogels. Besides, BGP promoted the early degradation and charring of KF-PVA matrix and formed a tubular structured reinforced char layer, which effectively played the shielding effect in condensed phase. Therefore, KF-PVA-BGP composites exhibited excellent fire safety properties. Besides, KF-PVA-BGP aerogel had good recyclability and biodegradability compared with commercial insulation materials. This multifunctional recyclable bio-based aerogel has exhibited tremendous potential for application in emerging fields.

Acknowledgements

This work was financially supported by the National Key R&D Program of China (2022YFC3080600 and 2022YFC3003105), Natural Science Foundation of China (52203082), Fundamental Research Funds for the Central Universities (2572022BU02), Key Research and Development Projects in Heilongjiang Province (GZ20220056) and Undergraduate Innovation Project for Northeast Forestry University (202110225445).

And the financial contributions from China Scholarship Council (202106600011) was also appreciated.

Highlights

• A 3D network structure of tubular aerogel was successfully constructed by the now mature freeze-drying technique.

• KF-PVA-BGP aerogels exhibited high strength and thermal insulation properties due to intermolecular hydrogen bonding.

• Compared to commercial petrochemical-based insulation materials, KF-PVA-BGP aerogels possessed excellent fire safety properties while maintaining its biodegradability and recyclability.

• BGP exerted its flame-retardant effect in KF aerogels through the synergistic effect of gas and condensed phase.

J o u r n a l P r e -p r o o f

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

J o u r n a l P r e -p r o o f

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