Fig 1.
(A) Mg2+ ions regulate MSC differentiation in both direct and indirect ways. (B) the cell model proposed in this study receives five cellular signals and predicts early and late differentiation rates. (C) the regulatory role of Mg2+ ions and the inflammatory cytokines on the early and late osteogenic differentiation. Upwards arrays in green indicate stimulatory roles; downwards arrays in red indicate inhibitory roles; up-down arrows in purple indicate dose-dependent effect. BioRender.com is used to create some elements of the graph.
Fig 2.
Fits of the model calibrated by C1-5 and C1 to the empirical data of study 1.
Bars indicate the simulations (S-) and the corresponding empirical data (E-) for increasing Mg2+ ion concentrations. The error bars on the empirical data show the standard deviations. The error bars on the simulation results show the standard deviations obtained during SSIP, i.e. 15% alteration in the estimated parameter values. Stars indicate the statistically significant differences between values given for the empirical data compared to the control, i.e. Mg2+ ion concentration of 0.08 mM (p < 0.05 = *; p < 0.01 = **). is the average fitness value of the simulations for the given measurement item.
Fig 3.
Fits of the model calibrated by C1-5 and C2 to the empirical data of study 2.
Bars indicate the simulations (S-) and the corresponding empirical data (E-) for increasing Mg2+ ion concentrations. The quantities of ALP and OC are reported at day 7 and 21, repectively. The error bars on the empirical data shows the standard deviations. The error bars on the simulation results show the standard deviations obtained during SSIP, i.e. 15% alteration in the estimated parameter values. Stars indicate the statistically significant differences between values given for the empirical data compared to the control, i.e. Mg2+ ion concentration of 0.8 mM (p < 0.05 = *). is the average fitness value of the simulations for the given measurement item.
Fig 4.
Fits of the model calibrated by C1-5 and C3 to the empirical data of study 3 for the case of IL-10.
Bars indicate the simulations (S) and the corresponding empirical data (E). The quantities of ALP and ARS are reported at day 14 and 21, respectively. The error bars on the empirical data shows the standard deviations. The error bars on the simulation results show the standard deviations obtained during SSIP, i.e. 15% alteration in the estimated parameter values. Stars indicate the statistically significant differences between values given for the empirical data compared to the control, i.e. the applied concentration of 0 ng/ml (p < 0.05 = *). is the average fitness value of the simulations for the given measurement item.
Fig 5.
Fits of the model calibrated by C1-5 and C3 to the empirical data of study 3 for the case of TNF-α.
Bars indicate the simulations (S) and the corresponding empirical data (E). The quantities of ALP and ARS are reported at day 14 and 21, respectively. The error bars on the empirical data shows the standard deviations. The error bars on the simulation results show the standard deviations obtained during SSIP, i.e. 15% alteration in the estimated parameter values. Stars indicate the statistically significant differences between values given for the empirical data compared to the control, i.e. the applied concentration of 0 ng/ml (p < 0.05 = *). is the average fitness value of the simulations for the given measurement item.
Fig 6.
Fits of the model calibrated by C1-5 and C4 to the empirical data of study 4.
Bars indicate the simulations (S) and the corresponding empirical data (E). The quantities of ALP and ARS are reported at day 3 and 9, respectively. The data is presented in a relative fold compared to the control, i.e. the undifferentiated case encoded as ctr. The error bars on the empirical data shows the standard deviations. The error bars on the simulation results show the standard deviations obtained during SSIP, i.e. 15% alteration in the estimated parameter values. Stars indicate the statistically significant differences between values given for the empirical data compared to the control (p < 0.05 = *). is the average fitness value of the simulations for the given measurement item.
Fig 7.
Fits of the model calibrated by C1-5 and C5 to the empirical data of study 5.
Bars indicate the simulations (S) and the corresponding empirical data (E). The quantities of ALP are reported at day 9. The error bars on the empirical data shows the standard deviations. The error bars on the simulation results show the standard deviations obtained during SSIP, i.e. 15% alteration in the estimated parameter values. Stars indicate the statistically significant differences between values given for the empirical data compared to the control, i.e. the applied concentration of 0 ng/ml (p < 0.05 = *; p < 0.01 = **). is the average fitness value of the simulations for the given measurement item.
Fig 8.
(A) Dispersity of the inferred values obtained during different runs of C1-5. In total, the calibration process is repeated 200 times in order to reach the stable inferred values, which is achieved by overlapping the mean values of all runs with the mean values of the 1st and 2nd halves of all runs. (B) Dispersity of the parameter values obtained during different calibration scenarios of C1, C2, C3, C4, C5, and C1-5. The values were scaled by dividing by the length of the priors. ED and LD stand for early differentiation and late differentiation, respectively.
Fig 9.
Results of the sensitivity analysis obtained during different calibration scenarios of C1, C2, C3, C4, C5 and C1-5.
The bars indicate the five most significant parameters, given on the scale of 1 to 5, obtained from LSSP. The numbers show the results of SSIP in percentage. Those parameters with no values were not involved in that particular study.
Table 1.
Fuzzy logic rules define the osteogenic reaction in response to stimulatory inputs.
For the qualitative definition of each linguistic term, refer to Fig 10A. The symbol ⊕ indicates that the given conditions must occur simultaneously to produce the given intensity. The symbol ~ indicates that any choice of one or more from the chosen inputs produces the same intensity. The term ‘Not’, preceded by a linguistic level, indicates that the rule applied for all except the given level. The terms ‘ED’ and ‘LD’ stand for early differentiation and late differentiation, respectively. The rules are given in IF/THEN format in S2 Table in.
Fig 10.
The complete flow of the implemented FL controller in this study.
In step 1, the cellular inputs are transformed to linguistic variables using the membership functions given in (A). A set of triangular and trapezoid memberships functions are used during this fuzzification process. Similarly, cellular outputs of early and late differentiation rates are defined in linguistic formats using the membership functions given in (B). A set of Gaussian membership functions are used for this purpose. In step 2, the FL controller receives the cellular inputs and calculates the cellular outputs, both in linguistic form. In step 3, the outputs are defuzzified and converted into crisp, real values using the centroid approach (C). The terms ‘ED’ and ‘LD’ stand for early differentiation and late differentiation, respectively.
Fig 11.
The complete flow of the calibration process and the sensitivity analysis implemenetd in this study.