Stability analysis of the deterministic model
Solving (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.
Growth 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.
The 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)}<0) (proof is in the supplementary information).
The cell density at the maximum RPR measures the fitness of the cell population and its proliferativeness. This cell density provides the idea about the extension of the proliferation phase, which is vital to predict the duration of culture and the number of observations to be taken during data collection.
Remark 4
The parameter (alpha) is the regulator of intercellular interactions/cell-cell cooperation. The parameter (delta) represents the regulator of interaction between cells and growth-inhibiting molecules to produce negative feedback. The equality of these two physical parameters indicates that the amount of signal a cell receives for proliferation is exactly the same as the signal of growth inhibition. In such a case, the environmental resistance and crowding regulator (beta) plays a key role in controlling the cell proliferation dynamics. The concept of conditional threshold cell density is not at all meaningful as the intercellular interactions/cell-cell cooperation regulator already reaches the value of growth inhibition regulation when (alpha =delta). This concept can be well understood through the usual stability analysis as described below. It is worthy of mentioning that mathematically we have only two equilibrium points, viz., one stable and another unstable, when (alpha =delta). For (alpha =delta) the proposed deterministic model (2) 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)}). This model has two equilibrium points with zero as the unstable equilibrium point and (K(1-frac{n}{r_{p}})^frac{1}{beta }) as the stable equilibrium point under the condition (r_{p}>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 estimation
The 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).
Our proposed model best fitted the cell density-RPR datasets for all of the seeding conditions generated through the grid-search method.
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.
Figure 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.
Time series solution of the proposed model and logistic law with comparative RSS for all three seeding conditions.
Trends in cell densities under deterministic set up
The (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.
The distribution of conditional MSSCD and cell density at maximum RPR in n-K parametric plane.
The 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).
The 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.
The 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.
The 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.
Stochastic model analysis
Our 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.
The 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.
The 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).
Figures 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.
The 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).
Due 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.
Distribution 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).
The 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).
Distribution 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).
Distribution 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).
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)}<0).
Remark 6
It is meaningful to understand the mathematical difficulties involved with colored noise scenario. To observe the effect analytically, we must know the value of (f^{prime }(x)) at the conditional MSSCD (x(t)=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)) to find the functional form of the noise induced drift A(x) and the noise induced diffusion coefficient B(x) of the Fokker-Planck equation as depicted in the stochastic model analysis section. After finding these functional forms, one can similarly observe the effect of the Gaussian colored noise like the white noise. However, for the proposed deterministic model (2), observation of the colored noise is quite difficult. Therefore, for simplicity, we discuss the effect of the Gaussian white noise on cell proliferation and avoid the effect of the correlation times.
Source: Ecology - nature.com
