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Stability analysis of the deterministic modelSolving (left( x(t) times left( r_{p}x(t)^{(alpha )}left( 1-big (frac{x(t)}{K}big )^{beta }right) – nx(t)^{(delta )} right) right) =0), we obtain two stable and one unstable equilibrium points for the model. One stable equilibrium is trivial, i.e., (x(t)=0), another stable equilibrium point being the non-zero satisfying (left( r_{p}x(t)^{(alpha )}left( 1-big (frac{x(t)}{K}big )^{beta }right) – nx(t)^{(delta )} right) =0). Figure 1a shows three different equilibrium points of the model. In addition to the equilibrium, the model has two inflection points (Fig. 1a). At these inflection points the absolute growth rates are minimum and maximum. The density vs relative proliferation rate (RPR) profile of the model shows that the model can attain negative RPR for a positive cell density, suggesting that the model can portray the Allee phenomenon (Fig. 1b). Figure 1c,d portray the proliferation and decay phases, respectively through the model.Figure 1Growth dynamics of the proposed model: (a) Absolute proliferation rate (APR) profile considering (r_{p}=0.13), (K=1.43), (n=0.0095), (alpha =1.15), (beta =0.99) and (delta =0.2); (b) RPR profiles for different n and other same constant model parameters; (c) Cell population survive for (r_{p}=0.13), (K=1.43), (n=0.0095), (alpha =1.15), (beta =0.99) and (delta =0.2) with the initial cell density 0.1; (d) The population goes to extinction for the initial cell density 0.06 with the same constant parameters.Full size imageThe solution of the deterministic model finally provides two theorems.
Theorem 1

(x^{*}approx K -Kleft( frac{Big (beta r_{p}K^{alpha }+n delta K^{delta }Big )-sqrt{Big (beta r_{p}K^{alpha }+n delta K^{delta }Big )^{2}-2 left( 2 alpha beta r_{p}K^{alpha } +beta (beta -1)r_{p}K^{alpha }+delta (delta -1)nK^{delta } right) nK^{delta }}}{left( 2 alpha beta r_{p}K^{alpha } +beta (beta -1)r_{p}K^{alpha }+delta (delta -1)nK^{delta } right) }right)) is the conditional MSSCD for the intercellular-interaction-induced proliferative cells. The conditional threshold density for cell-proliferation upon interaction is (x^{*}=K -Kleft( frac{Big (beta r_{p}K^{alpha }+n delta K^{delta }Big )+sqrt{Big (beta r_{p}K^{alpha }+n delta K^{delta }Big )^{2}-2 left( 2 alpha beta r_{p}K^{alpha } +beta (beta -1)r_{p}K^{alpha }+delta (delta -1)nK^{delta } right) nK^{delta }}}{left( 2 alpha beta r_{p}K^{alpha } +beta (beta -1)r_{p}K^{alpha }+delta (delta -1)nK^{delta } right) }right)) (proof is in the supplementary information).
Allee and cooperation models are the only extended logistic law other than our model to provide a threshold population size for growth or proliferation. Our proposed model is superior to the Allee and cooperation model as it can detect the conditional threshold cell density for proliferation and regulate the density by its different parameters. For example, One may reduce the conditional threshold density by either regulating the interaction between growth-inhibiting molecules and cells ((delta)) or reducing the inhibiting molecule concentration (n).The conditional MSSCD from Theorem 1 is lower than the carrying capacity of the conventional logistic model due to growth-inhibiting molecules; it provides the expected cell density during culture in a given environment. Theorem 1 also states the set of parameters to control the cell proliferation and get the desired density during such cell cultures. A further question arises knowing this set of parameters: which one of the parameters in the expression is crucial in terms of application purpose? Since the (r_{p}) is the constant proliferation rate for a given cell line, controlling the conditional MSSCD is not possible through (r_{p}). We simulate the distribution of conditional MSSCD for other parametric planes to answer this question. For this, we use the parameter values obtained from the data.

Theorem 2

The RPR is maximum at the cell density (x^{*}= K-Kleft( frac{r_{p}beta K^{alpha -1}+ndelta K^{delta -1}}{2r_{p}alpha beta K^{alpha -1}+r_{p}beta (beta -1)K^{alpha -1}+ndelta (delta -1)K^{delta -1}}right)) for the concave downward profile under the condition (r_{p}alpha (alpha -1){x^{*}}{}^{(alpha -2)}-frac{r_{p}}{K^{beta }}(alpha +beta )(alpha +beta -1){x^{*}}{}^{(alpha +beta -2)}-ndelta (delta -1){x^{*}}{}^{(delta -2)}n) (see the supplementary information). The cell population sustain with any positive initial cell density x(t) and try to stabilize at (x(t)= K(1-frac{n}{r_{p}})^frac{1}{beta }). Therefore, bimodality vanishes and unimodality is observed for the case (alpha =delta) (r_{p} >n). The RPR profile will be concave downward always with the maximum RPR value is at the inflection point (x(t)= K(frac{(r_{p}-n)alpha }{r_{p}(alpha +beta )})^frac{1}{beta }). The deterministic potential function in this case is (U(x)=-Big [(r_{p}-n)frac{x^{(alpha +2)}}{(alpha +2)}-frac{r_{p}}{K^{beta }}frac{x^{(alpha +beta +2)}}{(alpha +beta +2)} Big ]). The minima of this effective potential function will be at (x(t)= K(1-frac{n}{r_{p}})^frac{1}{beta }) which is the maximum stable cell density for (r_{p} >n).
Parameter estimationThe density-RPR and time-density fitting to the scratch assay datasets show a lower RSS for our model than the logistic one for each of the three seeding conditions. The estimated parameters from the RPR fitting through the grid-search are in Table 2. Although the RSS for the RPR fitting of the seeding 2 is very low, the data itself is too scattered in both the upper and lower range for the small cell density. Therefore, there is a chance that regardless of the low RSS value, the fitting for seeding 2 may not reflect the actual estimates of the parameters with the bias in the data set (Fig. 2b). Nevertheless, the density-RPR fittings to the other two seeding density datasets do not suffer from bias (Fig. 2a,c).Table 2 Estimated model parameters from density-RPR fitting of our model.Full size table
Figure 2Our proposed model best fitted the cell density-RPR datasets for all of the seeding conditions generated through the grid-search method.Full size image
Jin et al.1 suggested that their two phase logistic model may share similarities with the Allee effect. However, they did not fit the Allee model stating seeding 2 and 3 were large enough seeding densities. We calculated the conditional threshold density, conditional MSSCD, density at the minimum and maximum RPR for the model from our estimated parameters (Table 3). The conditional threshold cell density calculated from our estimated parameters confirms that the smallest initial seeding density of the dataset was greater than the conditional threshold cell density.Table 3 Calculated cell densities from estimated parameters from our model fitting.Full size tableFigure 3 compares the portrayal of the data through our model with the fitting by Jin et al.1. The blue dashed line is the time-series fitting of the proposed model, and the red-colored line is the time-series fitting of the logistic model to the scratch assay data sets in the Fig. 3. The carrying capacity values are unexpectedly very high in the logistic fit, keeping the model near the exponential phase for the entire dataset. Thus the overall and two phase logistic fits are unrealistic compared to the highest cell density observed in the assay. Also, logistic fitting of the RPR profiles to the data after 18 h does not capture the whole scenario. The green solid and the violet dashed line represent the logistic time-density fit after and before 18 h density profiles respectively. The orange-colored lines in the Fig. 3 are the expected population density as per estimated parameters from the RPR fitting after 18 h data sets. Table 4 enlists all parameters for a comparison between logistic and our model fitting.Figure 3Time series solution of the proposed model and logistic law with comparative RSS for all three seeding conditions.Full size imageTable 4 Logistic model fitting with the Jin et al.1 estimates used in Fig. 3 with the specific colors.Full size tableTrends in cell densities under deterministic set upThe (r_{p}) is fixed for a cell line among all the determining parameters of the conditional MSSCD. n and K vary together with the culture media, flask, and environmental setup. On the other hand, the (alpha), (beta), and (delta) vary together with intercellular-interactions and cellular-interaction with growth-inhibitory molecules, which depend on the medium’s initial cell density per well and fluidity. We observe that the distribution of the conditional MSSCD depends more on the K than the n. There is a chance of overproliferation in the deterministic setup under low n but high K. The cells may die under high n. The cell density at maximum RPR also depends more on K than n (Fig. 4). So the cells should be cultured in the larger flask to achieve maximum proliferativeness.Figure 4The distribution of conditional MSSCD and cell density at maximum RPR in n-K parametric plane.Full size imageThe conditional MSSCD depends more on (beta) than (alpha) (Fig. 5a). The cells may tend to overproliferate under both high (alpha) and (beta). The conditional MSSCD does not exist for a high (delta) and low (beta) depending more on (delta) than (beta). The cells may overproliferate only under a high (beta) and low (delta) (Fig. 5b). The conditional MSSCD also depends more on (delta) than (alpha) showing mostly underproliferation of cells in the (delta ~-alpha) parametric plane. Therefore, the proliferation can be controlled via regulating the interaction between the growth-inhibitory molecules and cells followed by density-regulation through contact-inhibition and cell-cell cooperation (Fig. 5c).Figure 5The distribution of the conditional MSSCD in parametric plane of regulators in the growth law: (a) dependence of the conditional MSSCD on (alpha) and (beta) parameters; (b) dependence of the conditional MSSCD on (delta) and (beta) parameters; (c) dependence of the conditional MSSCD on (alpha) and (delta) parameters.Full size imageThe new cell fitness measure, i.e. cell density at maximum RPR depends more on the (alpha) than the (beta) (Fig. 6a). The cells achieve maximum RPR at a great cell density under the high value of these two parameters. Figure 6b,c suggest that cell density depends only a little on the (delta) under high (alpha) and (beta). Under the low value of these two regulators, a high (delta) always reduces the cell density attaining the maximum RPR, resulting a poor cell-fitness.Figure 6The distribution of cell density at maximum RPR in parametric plane of regulators in the growth law: (a) dependence on (alpha) and (beta) parameters; (b) dependence on (alpha) and (delta) parameters; (c) dependence on (delta) and (beta) parameters.Full size imageStochastic model analysisOur proposed stochastic model (3) can be compared with the general stratonovich stochastic differential equation (frac{dx}{dt}=f(x)+g_{1}(x)epsilon (t)+g_{2}(x)Gamma (t)). Comparing it with our proposed stochastic model we obtain (g_{1}(x)=-x^{delta +1}) and (g_{2}(x)=1). Using the help of47, we get noise induced drift (A(x)=r_{p}x^{alpha +1}left( 1-Big (frac{x}{K}Big )^{beta } right) -nx^{(delta +1)}+D(delta +1)x^{(2delta +1)}-lambda sqrt{DQ}(delta +1)x^{delta }) and noise induced diffusion coefficient (B(x)=Dx^{(2delta +2)}-2lambda sqrt{DQ}x^{(delta +1)}+Q). The cell density at long run can be obtained from the steady state probability density function (SSPDF). The analytical expression of the SSPDF is obtained from the Fokker-Planck equation. The Fokker-Planck equation is (frac{partial P(x, t)}{partial t} =- frac{partial big [ A(x) P(x, t)big ]}{partial x}+ frac{partial ^{2} big [B(x) P(x, t)big ]}{partial x^{2}}), where P(x,t) is the probability density function of the cell population at the time point t. Solving the Fokker-Planck equation we get the SSPDF as (P_{st} (x)= frac{N^{prime }}{B(x)} exp left( int _{x} frac{A(x^{prime })}{B(x^{prime })} dx^{prime }right)) with the normalizing constant (N^{prime }). The value of (N^{prime }) can be obtained from (int _{0}^{infty } P_{st} (x)dx=1).This SSPDF (P_{st} (x)) helps to understand the validity of the proposed stochastic model. Since the number of the data points is too low to fit the stochastic model to the data directly, validation of the stochastic model is challenging in this case. The dataset we used is a time series with 15 data points with three replicates only. An experiment must have many replicates to have a sample with a large sample size so that the SSPDF of cell densities obtained from theoretical findings can be validated with the real observation of cell densities at the steady state. Such datasets with many replicates are rare.So, we generate 2000 sample paths with the help of numerical simulation based on stochastic model 3. We use the parameter values estimated from the fittings of the deterministic model to the seeding condition 1, and we consider some particular values for the two noise intensities and correlation strength ((lambda)) to get a simulated dataset. To achieve the stationary state, we consider sufficiently large time points, and the cell densities at the final time point are used as the data set for the stationary state. We compare the frequency density of cell densities at steady-state of a simulated dataset of 2000 sample paths with the SSPDF obtained from the analytical solution. This comparison shows that the cell density distribution at the steady state matches the steady state probability density function obtained analytically (Fig. 7).In addition, we illustrated the time series generated with the help of stochastic model 3 through numerical technique (Fig. 8). We have plotted the time series data thus obtained for each of the three seeding conditions and in the same figure we also plotted the observed cell densities. The red dots (o) represent the original/experimental dataset of Jin et al.1. The blue dots ((*)) represent the simulated dataset obtained from the stochastic model. This Fig. 8 clarifies our claim that the proposed stochastic model is in good agreement with the actual observation.Figure 7The histogram shows the distribution of cell densities at steady state under additive and multiplicative noises. The blue curve is the SSPDF. The function SSPDF and the distribution of cell densities matches to each other.Full size imageFigure 8The red dots (o) in each sub-figures represent the experimental data of Jin et al.1. The blue dots ((*)) are obtained from the stochastic model (3) considering: (a) The seeding 1 estimated model parameters with (D= 0.002), (Q= 0.06) and (lambda = 0.4). (b) The seeding 2 estimated model parameters with (D= 0.01), (Q= 0.15) and (lambda = 0.6). (c) The seeding 3 estimated model parameters with (D= 0.002), (Q= 0.2) and (lambda = 0.4).Full size imageFigures 7 and 8 suggest that the stochastic model is valid. So the model can be further analyzed to meet the first objective. Differentiating (P_{st} (x)), we obtain (frac{dP_{st} (x)}{dx}=frac{N^{prime }}{[B(x)]^2} exp left( int frac{A(x)}{B(x)}dx right) left( A(x)-frac{dB(x)}{dx} right)) and (frac{d^{2}P_{st} (x)}{dx^{2}}= frac{N^{prime }}{[B(x)]^{2}}exp left( int frac{A(x)}{B(x)}dx right) left( frac{dA(x)}{dx}-frac{d^{2}B(x)}{dx^{2}} right) +frac{N^{prime }}{[B(x)]^{2}} left( A(x)-frac{dB(x)}{dx} right) exp left( int frac{A(x)}{B(x)}dx right) frac{A(x)}{B(x)}-frac{2}{[B(x)]^3}N^{prime } exp left( int frac{A(x)}{B(x)}dx right) left( A(x)-frac{dB(x)}{dx} right) frac{dB(x)}{dx}). At the extrema of the SSPDF, we must have (frac{dP_{st} (x)}{dx}=0) i.e. (left( A(x)-frac{dB(x)}{dx} right) =0).

Theorem 3

(x^{*}approx K-K left( frac{nK^{delta +1}+D(delta +1) K^{2delta +1}-lambda sqrt{DQ}(delta +1)K^{delta }}{beta r K^{alpha +1}+n(delta +1) K^{(delta +1)}+D(delta +1) (2delta +1)K^{(2delta +1)}-lambda sqrt{DQ}delta (delta +1)K^{delta }} right)) is the conditional MSSCD due to the correlated additive and multiplicative noises under the condition (r_{p}(alpha +1)x^{*}{}^{alpha }-frac{r_{p}}{K^{beta }}(alpha +beta +1)x^{*}{}^{(alpha +beta )} -n(delta +1)x^{*}{}^{delta }-D(delta +1)(2delta +1)x^{*}{}^{(2delta )}+lambda sqrt{Dalpha }delta (delta +1)x^{*}{}^{(delta -1)} < 0) (proof is in the supplementary information). Figure 9 visualizes the effect of noise strength and correlation strength on the conditional MSSCD. The conditional MSSCD increases with the additive noise strength (Q) and decreases with the multiplicative noise strength (D) when the other model parameters are fixed (Fig. 9a). There is a high chance of overproliferation for a low D and a high Q (Fig. 9a). Again, there is a high chance of extinction for the low Q and high D. The conditional MSSCD depends more on D than (lambda) (Fig. 9b), and more on (lambda) than Q (Fig. 9c). The conditional MSSCD increases with (lambda) and Q; there is a high chance of overproliferation for high (lambda) and Q. The extinction risk of cells from the culture increases with low (lambda) and Q.Figure 9The change in the conditional MSSCD value for different noise strengths and correlation strength using the parameters estimated for seeding 1: (a) the conditional MSSCD values in (D-Q) noise strength plane with highest correlation ((lambda =1)); (b) the conditional MSSCD values in (D-lambda) noise plane with (Q=0.01); (c) the conditional MSSCD values in (Q-lambda) noise plane with (D=0.01).Full size imageDue to the difficulty and complicated expression of the analytical expression of the SSPDF, we use numerical simulation to study the steady-state behavior in the long run under correlated noises. We draw a histogram of the cell densities based on 500 normal sample paths at the final time points. We use seeding 1 fitting estimates as the initial parameter values for this simulation. The cell population is stable and steady at either 0 cell density or at the conditional MSSCD. The distribution is symmetric around the conditional MSSCD for (lambda =1) (Fig. 10a). There is a loss in the symmetry for the decreasing (lambda). For (lambda =0.5), there is a mode at the zero states with another mode at conditional MSSCD (Fig. 10b). The histogram shows a bi-modality for low values of (lambda). The mode at the zero state is highest for (lambda =0) (Fig. 10c). Therefore, the extinction chance increases for zero noise correlation between the additive and the multiplicative noises.Figure 10Distribution of cell density for (r_{p}=0.13), (K=1.43), (n=0.0095), (alpha =1.15), (beta =0.99), (delta =0.2), (D=0.01), (Q=0.01), and variable correlation between additive and multiplicative noises: (a) (lambda =1), (b) (lambda =0.5) and (c) (lambda =0).Full size imageThe sustainability of the cell population depends on the strength of the two noises, like the correlation strength between them. For the zero strength multiplicative noise, the population has the mode at around the conditional MSSCD value (Fig. 11). Therefore, the population sustains in this case and tries to stabilize at the conditional MSSCD value. For (D=0.02), there is a bimodality, where the highest mode is at the zero cell density. For (D=0.05), we observe only one mode at (x=0). Therefore, with the increasing values of the multiplicative noise strengths (D), the chance of extinction increases for (lambda =0.5), (Q=0.01), and other constant model parameters for the seeding condition 1. Similar things happen for increasing Q values considering (D=0.01), (lambda =0.5), and other constant model parameters (Fig. 12).Figure 11Distribution of cell density for (r_{p}=0.13), (K=1.43), (n=0.0095), (alpha =1.15), (beta =0.99), (delta =0.2), (lambda =0.5), (Q=0.01), and variable strength of multiplicative noise: (a) (D=0.05), (b) (D=0.02) and (c) (D=0).Full size imageFigure 12Distribution of cell density for (r_{p}=0.13), (K=1.43), (n=0.0095), (alpha =1.15), (beta =0.99), (delta =0.2), (lambda =0.5), (D=0.01), and variable correlation between multiplicative noise: (a) (Q=0.05), (b) (Q=0.02) and (c) (Q=0).Full size image Remark 5 We have previously discussed the scenario for (alpha =delta) for deterministic case in Remark 4. It is important to understand the scenario under stochastic case too. For (alpha =delta) the proposed stochastic model 3 becomes (frac{dx(t)}{dt}=r_{p}x(t)^{(alpha +1)}left( 1-big (frac{x(t)}{K}big )^{beta }right) - nx(t)^{(alpha +1)}-x(t)^{(alpha +1)} epsilon (t)+ Gamma (t)). For this stochastic model (g_{1}(x)=-x^{alpha +1}) and (g_{2}(x)=1). We get, (A(x)=r_{p}x^{alpha +1}left( 1-Big (frac{x}{K}Big )^{beta } right) -nx^{(alpha +1)}+D(alpha +1)x^{(2alpha +1)}-lambda sqrt{DQ}(alpha +1)x^{alpha }) and (B(x)=Dx^{(2alpha +2)}-2lambda sqrt{DQ}x^{(alpha +1)}+Q). The extrema of the SPDF (big (x(t)=x^{*}big )) must satisfy the growth equation (r_{p}{x^{*}}^{alpha +1}-frac{r_{p}}{K^{beta }}(x^{*})^{alpha +beta +1}-n(x^{*})^{alpha +1}-D(alpha +1)(x^{*})^{2alpha +1}+lambda sqrt{D~Q}(alpha +1)(x^{*})^{alpha }=0). Therefore, for (alpha =delta) the conditional MSSCD is (x^{*}= K-Kfrac{nK^{(alpha +1)}+D(alpha +1)K^{(2alpha +1)}-lambda sqrt{DQ}(alpha +1)K^{alpha }}{beta r_{p}K^{(alpha +1)}+nK^{(alpha +1)}(alpha +1)+D(alpha +1)(2alpha +1)K^{(2alpha +1)}-alpha lambda sqrt{DQ}(alpha +1)K^{alpha }}) under the condition ((r_{p}-n)(alpha +1)(x^{*})^{alpha }-frac{r_{p}}{K^{beta }}(alpha +beta +1)(x^{*})^{(alpha + beta )}-(alpha +1)(2alpha +1)D(x^{*})^{2alpha }+lambda sqrt{DQ}(alpha +1)alpha (x^{*})^{(alpha -1)} More

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Choice of biowaste treatment plants and sample identifiersCompost samples were collected from four central municipal biowaste treatment plants (denominated as #1 to #4) in Baden-Wurttemberg, Germany (Table 1). All plants used a state-of-the-art two-stage biowaste treatment process comprising of (a) anaerobic digestion/biogas production and (b) subsequent composting of the solid digestate to produce a high-quality mature compost sold for direct use as fertilizer in agriculture. The composts were regularly analyzed by an independent laboratory for quality and residual contamination and consistently fulfilled the quality requirements of the label RAL-GZ 251 Gütezeichen Kompost of the German Bundesgütegemeinschaft Kompost e.V. (www.gz-kompost.de). Plants #1 and #3 produce in addition a liquid fertilizer, which is separated from the solid digestate at the end of stage a) by press filtration and which is also intended for direct use on agricultural soil (replacement of liquid manure). In case of plants #1, #3, and #4 up to 25 wt% of shrub/tree cuttings were added to the solid digestate for composting. All plants used sieving (typically with a 12 or a 20 mm mesh) at the end of the process to assure the necessary purity of their finished composts. Whenever technically possible, we as well took samples of the pre-compost immediately before this final sieving step to evaluate its contribution to the removal of residual BPD fragments. For analysis, composts were passed consecutively through two sieves with mesh sizes of 5 mm and 1 mm, yielding two fragment preparations for IR-analysis namely a > 5 mm fraction corresponding to the contamination by residual “macroplastic” (5 mm is a commonly used upper size limit for “microplastic”, anything larger is macroplastic) and a 1–5 mm fraction corresponding to the regulatory relevant residual contamination by microplastic. The lower limit of 1 mm rather than 2 mm was chosen in anticipation of the expected changes in regulation, where the replacement of the 2 mm limit by a 1 mm limit is imminent.Table 1 Technical data of the investigated plants and incidence of BDP fragments in the sampled composts.Full size tableOccurrence of plastic fragments  > 1 mm in the sampled compostsComposting times of 5–9 weeks were used in the investigated plants (Table 1), which is shorter than the 12 weeks indicated in EN 13432 for the 90% disintegration of a compostable plastic material, but a realistic time span for state-of-the-art technical waste treatment. Since we were not in a position to estimate the quantity of BDP entering the plants, since for technical reasons we were unable to obtain a representative sample, we cannot say, whether any residual BDP detected by us in the finished composts was due to a yet incomplete disintegration process or whether it corresponds to the 10% material still permissible by EN 13432 even after the full composting step. However, in 7 out of the 12 sampled composts and pre-composts fragments with chemical signatures corresponding to the BDPs poly (lactic acid) (PLA) and poly (butylene-adipate-co-terephthalate) (PBAT) were identified in the > 5 mm and/or the 1–5 mm sieving fractions using FTIR analysis3 (Fig. 1; Table 1). All recovered fragments appeared to stem from foils, bags or packaging, since they were thin compared to their length and width (see Suppl Figure S1 for typical examples). Fragments with overlapping signatures, most likely PBAT/PLA mixtures or blends, were also found (see Suppl Figure S2 for the interpretation of the spectra). In addition, the recorded BDP fragment spectra (Fig. 1A) showed high similarity to the FTIR spectra of commercial compostable bags sold in the vicinity of the biowaste treatment plants (Fig. 1B), which together with the geometry of the recovered fragments led us to assuming that the majority of the BDP entered the biowaste in the form of such bags.Figure 1FTIR spectra of BDP fragments from composts and commercial bags. (A) BDP fragments recovered from the composts and (B) the commercial compostable bags. Fragments were coded as follows: p or f for pre-compost or finished compost, followed by the plant number (#1 to #4), an indication of the size fraction ( > 5 mm or 1–5 mm) in which the fragment was found, and finally, the fragment number. Fragment F#1_5mm_4 therefore represents the 4th fragment collected in the  > 5 mm size fraction from the finished compost of plant number 1. Bags were arbitrarily numbered 1–10, see Suppl Table S1 for supplier information. The spectra (in grey) of the reference materials for PLA and PBAT are given as basis for the interpretation. Spectra in red refer to test samples consisting only of PBAT, while those in blue indicate samples composed of PBAT/PLA mixtures.Full size imageThe BDP fragments were found alongside fragments of commodity plastics (mostly PE) in all cases. Finished composts tended to contain fewer and smaller fragments than the corresponding pre-composts. The final sieving of the pre-composts to prepare the finished composts hence appears to be quite effective in removing such fragments, in particular those from the > 5 mm size fraction (Table 1) and for that reason has become state-of-the-art in preparing quality composts (contamination by plastic fragments > 2 mm of less than 0.1 wt%). Given that the size of the fragments is a crucial factor regarding ecological risk, we analyzed the sizes (length Î width) of the BDP fragments in comparison to that of the plastic fragments with signatures of commodity plastics such as PE (Fig. 2). BDP fragments found in a given compost sample tended to be smaller than the fragments stemming from non-BDP materials, which may indicate that BDPs degrade faster or tend to disintegrate into tinier particles than commodity plastics. This may also explain why in the compost from plant #2, no BDP fragments were found in the particle fraction retained by the 5 mm sieve ( > 5 mm fraction), while 19 such particles were found in the fraction then retained by the 1 mm sieve (1–5 mm fraction). Interestingly, plant #2 is the only one included in our study that uses no mechanical breakdown of the incoming biowaste. This reduces the mechanical stress on the incoming material. Mechanical stress can alter the properties of plastic foils such as the crystallinity whereby crystallinity has been shown to influence the biological degradation of BDP such as PLA7.Figure 2Size distribution of plastic fragments  > 1 mm. (A) Fragments found in the finished compost from plant #1, (B) in the finished compost from plant #2, and (C) in the pre-compost from plant #3. For reasons of statistical relevance, only samples containing more than 20 BDP fragments per kg of compost were included in the analysis.Full size imageMaterial characteristics of BDP fragments in comparison to those of commercial biodegradable bagsIn order to verify whether the BDP fragments recovered from the composts differed from the compostable bags in any parameter with possible relevance for biodegradation and environmental impact16, the physico-chemical properties of bags and fragments were studied in detail. Since we wanted to have a maximum of information of the BDP fragments, size/weight was a limiting factor in selecting fragments for analysis. Fragments of at least 1 mg were required for the FT-IR analysis. 5 mg-fragments could be analyzed in addition by 1H-NMR, while the full set of analytics (FT-IR, 1H-NMR, and DSC) required at least 10 mg of sample.For insight into the chemical composition, 1H-NMR spectra of the commercial bags and all suitable BDP fragments were compared (Fig. 3). In case of material mixtures and blends, the 1H-NMR analysis allows quantification of the PBAT/PLA weight ratio in the materials and also of the ratio of the butylene terephthalate (BT) and butylene adipate (BA) units in the involved PBAT polyesters.Figure 31H NMR spectra of BDP fragments from composts and commercial bags. (A) BDP fragments recovered from the composts and (B) the commercial compostable bags. Fragments were coded as follows: p or f for pre-compost or finished compost, followed by the plant number (#1 to #4), an indication of the size fraction ( > 5 mm or 1–5 mm) in which the fragment was found, and finally, the fragment number. Bags were arbitrarily numbered 1–10, see Suppl Table S1 for supplier information. The spectra (in grey) of the reference materials for PLA and PBAT are given as basis for the interpretation. Spectra in red refer to test samples consisting only of PBAT, while those in blue indicate samples composed of PBAT/PLA mixtures. (C) Chemical structures of PLA and PBAT, chemical shifts of the protons are assigned as indicated in the reference spectra in (B).Full size imageThe 1H-NMR spectra corroborate the FTIR measurements in that all investigated commercial bags were made from PBAT/PLA mixtures of varied composition (Table 2). By comparison, some of the fragments, for instance, f#1_5mm_4, appeared to consist of only PBAT. Other fragments, e.g., f#1_1mm_9, were mixtures of PLA and PBAT (Table 2). However, even in the case of PBAT/PLA mixtures, the average PBAT content tended to be higher in the fragments than in the bags, while the BT/BA monomer ratio in the respective PBATs, was also significantly higher in the fragments than in the bags. If we assume the fragments to stem from similar compostable bags as the ones included in our comparison, this would mean that during composting of such a bag, the PLA degrades more quickly than the PBAT, whereas within a given PBAT polyester, the BA unit is more easily degraded than the BT unit. Evidence can indeed be found in the pertinent literature that PLA has faster biodegradation kinetics than PBAT, while BT is more resistant to mineralization than BA17,18.Table 2 Composition of commercial compostable bags and BDP fragments recovered from the composts as analyzed by 1H-NMR.Full size tableNext, differential scanning calorimetry (DSC) was used to analyze fragments compared to commercial bags in regard to the presence of amorphous vs. crystalline domains, a parameter expected to affect biodegradation kinetics and therefore the putative environmental impact of the produced microplastic16 upon release into the environment with the composts. Whereas amorphous domains show glass transition, crystalline domains show melting, both of which can be discerned by the respective phase transition enthalpy in the DSC curves (Fig. 4).Figure 4DSC curves of BDP fragments and compostable bags #1 and #7. Curves for the reference materials (in grey) for PLA and PBAT are given for comparison. Curves were recorded during the first heating run (temperature range: − 50 °C to 200 °C, heating rate: 10 °C min−1). (A) and (B) curves in red refer to test samples consisting only of PBAT, while those in blue indicate samples composed of PBAT/PLA mixtures. Fragments were coded as follows: p or f for pre-compost or finished compost, followed by the plant number (#1 to #4), an indication of the size fraction ( > 5 mm or 1–5 mm) in which the fragment was found, and finally, the fragment number.Full size imageThe curve for the reference PBAT shows a glass transition temperature (Tg) of − 29 °C and a broad melting range between 100 and 140 °C for the crystalline domains, while that of the PLA reference shows a glass transition temperature of 58 °C and a narrower melting peak between 144 °C and 162 °C. The curve for commercial bag #1, which had a comparatively high PLA content, shows a pronounced melting peak in the expected range; the same is the case for fragment p#3_5mm_1 and to a lesser extent for fragment p#3_5mm_9, two fragments, which also have high PLA contents. The DSC curves of the other fragments and bag #1 are undefined in comparison, which is due to their high PBAT content. According to the DSC curves, most of the investigated materials are semicrystalline, i.e., contain both amorphous (glass transition) and crystalline (melting) domains. However, the DCS data alone allow only a qualitative discussion of the differences between fragments and bags.To obtain quantitative data on the crystallinity differences, wide angle X-ray scattering (WAXS) spectra were recorded. WAXS requires fragments at least 3 cm long, which restricted the number of fragment samples to three, all of which were found in pre-compost samples. The corresponding curves are shown in Fig. 5A–C. The spectra of the commercial biodegradable bags are shown in Suppl Figure S3. Foils were in addition prepared by heat pressing from the reference materials for PLA and PBAT in order to include them into the WAXS measurements (Fig. 5D). While the foils produced from the PBAT reference material produced crystallinity peaks at 16.2°, 17.3°, 20.4°, 23.2°, and 24.8°, the foil prepared from the PLA reference material showed only an amorphous halo at 15.5° and 31.5°, which is in accordance with values published in the literature19. A more pronounced crystallinity peak was obtained in the case of an additionally annealed PLA foil.Figure 5WAXS curves with Lorenz fitting for (A) fragment p#3_5mm_1, (B) fragment p#3_5mm_9, and (C) fragment p#4_5mm_2. (D) WAXS curves for foils produced from the PBAT and PLA reference materials; the percent values indicate the crystallinity. The dash lines are the fitting peak curves for the XRD spectrum. Crystallinity can be obtained by dividing the integration area of the fitted peaks by the integration area of the entire spectrum. Fragments were coded as follows: p or f for pre-compost or finished compost, followed by the plant number (#1 to #4), an indication of the size fraction ( > 5 mm or 1–5 mm) in which the fragment was found, and finally, the fragment number.Full size imageIn case of the fragments and bags, the peaks of PLA and PBAT overlapped to some extent in the WAXS spectra, but by conducting Lorenz fitting using Origin software, the overall crystallinity could be calculated as follows:$$chi = { 1}00% , *{text{ Aa}}/left( {{text{Aa }} + {text{ Ac}}} right)$$where χ is the crystallinity and Aa and Ac represent the areas of the amorphous and crystalline peaks.Using this equation, crystallinities of 55% (fragments p#3_5mm_1), 34% (p#3_5mm_9), and 34% (p#4_5mm_2) were calculated for the fragments. The foils prepared in house for the reference materials had similar crystallinities (43% in case of the annealed PLA foil and 26% of the PBAT foil), while the simple PLA foil was amorphous. By comparison, for eight of the commercial bags, crystallinities in the range from 1% to 7% were calculated, whereas these values were 14% and 15% for the remaining two bag types (Suppl Figure S3).The high crystallinity of the larger fragments recovered from the pre-compost samples suggests that crystalline domains of BDP materials may indeed disintegrate more slowly than the amorphous ones, as prior studies on microbial biodegradation have suggested7,8. Admittedly, such large fragments per se would not enter the environment, since the final sieving step used to prepare the finished composts is quite efficient at removing them. However, it is tempting to extrapolate that residual BDP in general are remnants of the more crystal domains of the original material, even though experimental proof of this assumption is at present not possible. 10 wt% of a BDP bag is allowed to remain after standard composting. It is usually assumed that any such residues continue to degrade with comparable speed. However, should these residues correspond to the more crystalline domains, rather than degrading with similar speed as the bulk material, the more crystalline fragments can be expected to persist for a much longer and at present unpredictable length of time in the environment, e.g. when applied to the soil with the composts; in particular, when they are also enriched in PBAT and BT units as suggested by our analysis of the chemical composition. Data from the use of biodegradable foils in agriculture show that the degradation in the environment may take years20. Altogether this may have unforeseen economic and environmental consequences, especially when considering the high fraction of BDP fragments < 5 mm. Putative consequences include changes in soil properties, the soil microbiome and therefore in plant performance21, a factor indispensable for worldwide nutrition.Residues of BDP fragments  1 mm were found in the collected LF samples. This is hardly surprising, given that the LF is produced by press filtration of the digestate after the anaerobic stage. Such a filtration step can be expected to retain fragments > 1 mm in the produced filter cake, which goes into the composting step, leaving the filtrate, i.e. the LF, essentially free of such particles. Anaerobic digestion is currently not assumed to contribute significantly to the degradation of BDP17,22, but the process conditions (mixing, pumping) may promote breakdown of larger fragments, particularly when additives such as plasticizers23 leach out of the material.Since the residual solids content of the LF is low (plant #1: 8.6 wt%, plant #3: 5.8 wt%), a combination of enzymatic-oxidative treatment and µFTIR imaging originally developed for environmental samples from aqueous systems24,25 could be adapted for the analysis (size and chemical signature) of particles in the LF down to a size of 10 µm. The corresponding data are compiled in Table 3. In all cases, residual fragments from PBAT-based polymers represented the dominant plastic fraction in the investigated samples; i.e. approximately 53% of all plastic particles in the LF from plant #1 (11,520 BDP particles per liter) and 65% in the case of plant #3 (12,480 BDP particles per liter). Liquid manure is applied several times a year to fields at a concentration of 2–3 L m−2. According to our analysis > 20,000 BDP microparticles of a size ranging from 10 µm to 500 µm enter each m2 of agricultural soil whenever LF is applied on agricultural surfaces.Table 3 Microplastic fragments (BDP/all) found per liter of liquid fertilizer.Full size tableDue to the complexity of the matrix, a similar analysis of individual plastic fragments  1 mm. Six compost samples representing the more contaminated ones based on the content of fragments > 1 mm, namely, f#1, f#2, p#3, f#3, p#4 and f#4 (nomenclature: f or p for finished or pre-compost, followed by plant number), were extracted with a 90/10 vol% chloroform/methanol mixture. The amounts of PBAT and PLA in the obtained extracts were then quantified via 1H-NMR (Table 4). Briefly, the intensity of characteristic signals in the extract spectra of the compost samples (see Suppl Figure S4) were compared to peak intensities produced by calibration standards of the pure polymer dissolved at a known concentration in the chloroform/methanol. All samples and standards were normalized using the 1,2-dichloroethan signal at 3.73 ppm as internal standard. See also Suppl Figure S5 for an exemplification of the quantification of the PBAT/PLA ratios. Based on the amounts of PBAT and PLA extracted from a known amount of compost, the total mass concentration (wt% dry weight) of these polymers in the composts was calculated.Table 4 Evidence of PBAT and PLA residues caused by fragments  2 mm. Moreover, residues of PBAT and PLA were found in all investigated compost samples, including the finished compost from plant #4, which had shown no contamination by larger BPD fragments (Table 1). The pre-compost from that plant had shown a few contaminating BDP fragments in the > 5 mm fraction. However, in regard to the fragments More