An in silico erythropoiesis model rationalizing synergism between stem cell factor and erythropoietin

Abstract

Stem cell factor (SCF) and erythropoietin (EPO) are two most recognized growth factors that play in concert to control in vitro erythropoiesis. However, exact mechanisms underlying the interplay of these growth factors in vitro remain unclear. We developed a mathematical model to study co-signaling effects of SCF and EPO utilizing the ERK1/2 and GATA-1 pathways (activated by SCF and EPO) that drive the proliferation and differentiation of erythroid progenitors. The model was simplified and formulated based on three key features: synergistic contribution of SCF and EPO on ERK1/2 activation, positive feedback effects on proliferation and differentiation, and cross-inhibition effects of activated ERK1/2 and GATA-1. The model characteristics were developed to correspond with biological observations made known thus far. Our simulation suggested that activated GATA-1 has a more dominant cross-inhibition effect and stronger positive feedback response on differentiation than the proliferation pathway, while SCF contributed more to the activation of ERK1/2 than EPO. A sensitivity analysis performed to gauge the dynamics of the system was able to identify the most sensitive model parameters and illustrated a contribution of transient activity in EPO ligand to growth factor synergism. Based on theoretical arguments, we have successfully developed a model that can simulate growth factor synergism observed in vitro for erythropoiesis. This hypothesized model can be applied to further computational studies in biological systems where synergistic effects of two ligands are seen.

Appendices

Appendix 1: model equations

Reactions for the model

From the left hand side of the proposed model schematic (Fig. 2), the following reactions are associated with cell proliferation.

$$\begin{aligned} & \to R_{\text{a}}R1_{\text{a}} = k1 \\ & R_{\text{a}} \to R2_{\text{a}} = k2*R_{\text{a}} \\ & R_{\text{a}} \to C_{\text{a}} R3_{\text{a}} = k3*L_{\text{a}} *R_{\text{a}} \\ & C_{\text{a}} \to R_{\text{a}} R4_{\text{a}} = k4*C_{\text{a}} \\ & C_{\text{a}} \to R5_{\text{a}} = k5*C_{\text{a}} \\ & \to {\text{IF}}_{\text{a}}R6_{\text{a}} = k6 \\ & {\text{IF}}_{\text{a}} \to R7_{\text{a}} = k7*{\text{IF}}_{\text{a}} \\ & {\text{IF}}_{\text{a}} \to {\text{AF}}_{\text{a}} \\ & R8_{\text{a}} = \frac{{k8*{\text{IF}}_{\text{a}} *(k_{\text{a}} *C_{\text{a}} + k_{\text{b}} *C_{\text{b}} )}}{{k81 + {\text{IF}}_{\text{a}} }}\\ & \qquad *\frac{{\left( {k_{\text{a}} *C_{\text{a}} + k_{\text{b}} *C_{\text{b}} } \right)}}{{\left( {k_{\text{a}} *C_{\text{a}} + k_{\text{b}} *C_{\text{b}} } \right) + K}}*\frac{{L_{\text{a}}^{ 3} }}{{L_{\text{a}}^{ 3} + 100^{3} }}*\frac{{L_{\text{b}}^{ 3} }}{{L_{\text{b}}^{ 3} + 100^{3} }} \\ & {\text{AF}}_{\text{a}} \to {\text{IF}}_{\text{a}}R9_{\text{a}} = \frac{{k9*P_{\text{a}} *{\text{AF}}_{\text{a}} }}{{k91 + {\text{AF}}_{\text{a}} }} \\ & {\text{AF}}_{\text{a}} \to R10_{\text{a}} = k10*{\text{AF}}_{\text{a}} \\ & \to R_{\text{a}}R11_{\text{a}} = \frac{{k11*{\text{AF}}_{\text{a}} }}{{k111*\left( {1 + \frac{{{\text{AF}}_{\text{b}} }}{kI}} \right) + {\text{AF}}_{\text{a}} }} \\ & \to {\text{IF}}_{\text{a}}R12_{\text{a}} = \frac{{k12*{\text{AF}}_{\text{a}} }}{{k121*\left( {1 + \frac{{{\text{AF}}_{\text{b}} }}{kI}} \right) + {\text{AF}}_{\text{a}} }} .\\ \end{aligned}$$

From the right hand side of the proposed model schematic (Fig. 2), the following reactions are associated with cell differentiation.

$$ \begin{aligned} & \to R_{\text{b}} R1_{\text{b}} = k\_1 \\ & R_{\text{b}} \to R2_{\text{b}} = k\_2*R_{\text{b}} \\ & R_{\text{b}} \to C_{\text{b}} R3_{\text{b}} = k\_3*L_{\text{b}} *R_{\text{b}} \\ & C_{\text{b}} \to R_{\text{b}} R4_{\text{b}} = k\_4*C_{\text{b}} \\ & C_{\text{b}} \to R5_{\text{b}} = k\_5*C_{\text{b}} \\ & \to {\text{IF}}_{\text{b}} R6_{\text{b}} = k\_6 \\ & {\text{IF}}_{\text{b}} \to R7_{\text{b}} = k\_7*{\text{IF}}_{\text{b}} \\ & {\text{IF}}_{\text{b}} \to {\text{AF}}_{\text{b}} R8_{\text{b}} = \frac{{k\_8*IF_{\text{b}} *C_{\text{b}} }}{{k\_81 + {\text{IF}}_{\text{b}} }} \\ & {\text{AF}}_{\text{b}} \to {\text{IF}}_{\text{b}} R9_{\text{b}} = \frac{{k\_9*P_{\text{b}} *{\text{AF}}_{\text{b}} }}{{k\_91 + {\text{AF}}_{\text{b}} }} \\ & {\text{AF}}_{\text{b}} \to R10_{\text{b}} = k\_10*{\text{AF}}_{\text{b}} \\ & \to R_{\text{b}} R11_{\text{b}} = \frac{{k\_11*{\text{AF}}_{\text{b}} }}{{k\_111*\left( {1 + \frac{{{\text{AF}}_{\text{a}} }}{k\_I}} \right) + {\text{AF}}_{\text{b}} }} \\ & \to {\text{IF}}_{\text{b}} R12_{\text{b}} = \frac{{k\_12*{\text{AF}}_{\text{b}} }}{{k\_121*\left( {1 + \frac{{{\text{AF}}_{\text{a}} }}{k\_I}} \right) + {\text{AF}}_{\text{b}} }}. \\ \end{aligned}$$

Differential equations for the model

$$\begin{aligned} \frac{{d(R_{\text{a}} )}}{dt} & = k1 - k2*R_{\text{a}} - k3*L_{\text{a}} *R_{\text{a}} + k4*C_{\text{a}} + \frac{{k11*{\text{AF}}_{\text{a}} }}{{k111*\left( {1 + \frac{{{\text{AF}}_{\text{b}} }}{kI}} \right) + {\text{AF}}_{\text{a}} }} \\ \frac{{d(C_{\text{a}} )}}{dt} & = k3*L_{\text{a}} *R_{\text{a}} - k4*C_{\text{a}} - k5*C_{\text{a}} \\ \frac{{d({\text{IF}}_{\text{a}} )}}{dt} & = k6 - k7*{\text{IF}}_{\text{a}} - \frac{{k8*{\text{IF}}_{\text{a}} *\left( {k_{\text{a}} *C_{\text{a}} + k_{\text{b}} *C_{\text{b}} } \right)}}{{k81 + {\text{IF}}_{\text{a}} }}*\frac{{\left( {k_{\text{a}} *C_{\text{a}} + k_{\text{b}} *C_{\text{b}} } \right)}}{{\left( {k_{\text{a}} *C_{\text{a}} + k_{\text{b}} *C_{\text{b}} } \right) + K}}*\frac{{L_{\text{a}}^{3} }}{{L_{\text{a}}^{3} + 100^{3} }}*\frac{{L_{\text{b}}^{3} }}{{L_{\text{b}}^{3} + 100^{3} }} \\ & \quad+ \frac{{k9*P_{\text{a}} *{\text{AF}}_{\text{a}} }}{{k91 + {\text{AF}}_{\text{a}} }} + \frac{{k\_12*{\text{AF}}_{\text{a}} }}{{k\_121*\left( {1 + \frac{{{\text{AF}}_{\text{b}} }}{k\_I}} \right) + {\text{AF}}_{\text{a}} }} \\ \frac{{d({\text{AF}}_{\text{a}} )}}{dt} & = \frac{{k8*{\text{IF}}_{\text{a}} *\left( {k_{\text{a}} *C_{\text{a}} + k_{\text{b}} *C_{\text{b}} } \right)}}{{k81 + IF_{\text{a}} }}*\frac{{\left( {k_{\text{a}} *C_{\text{a}} + k_{\text{b}} *C_{\text{b}} } \right)}}{{\left( {k_{\text{a}} *C_{\text{a}} + k_{\text{b}} *C_{\text{b}} } \right) + K}}*\frac{{L_{\text{a}}^{3} }}{{L_{\text{a}}^{ 3} + 100^{3} }}*\frac{{L_{\text{b}}^{ 3} }}{{L_{\text{b}}^{3} + 100^{3} }} - \frac{{k9*P_{\text{a}} *{\text{AF}}_{\text{a}} }}{{k91 + {\text{AF}}_{\text{a}} }} - k10*{\text{AF}}_{\text{a}} \\ \frac{{d(R_{\text{b}} )}}{dt} & = k\_1 - k\_2*R_{\text{b}} - k\_3*L_{\text{b}} *R_{\text{b}} + k\_4*C_{\text{b}} + \frac{{k\_11*{\text{AF}}_{\text{b}} }}{{k\_111*\left( {1 + \frac{{{\text{AF}}_{\text{a}} }}{k\_I}} \right){\text{ + AF}}_{\text{b}} }} \\ \frac{{d(C_{\text{b}} )}}{dt} & = k\_3*L_{\text{b}} *R_{\text{b}} - k\_4*C_{\text{b}} - k\_5*C_{\text{b}} \\ \frac{{d({\text{IF}}_{\text{b}} )}}{dt} & = k\_6 - k\_7*{\text{IF}}_{\text{b}} - \frac{{k\_8*{\text{IF}}_{\text{b}} *C_{\text{b}} }}{{k\_81 + {\text{IF}}_{\text{b}} }} + \frac{{k\_9*P_{\text{b}} *{\text{AF}}_{\text{b}} }}{{k\_91{\text{ + AF}}_{\text{b}} }} + \frac{{k\_12*{\text{AF}}_{\text{b}} }}{{k\_121*\left( {1 + \frac{{{\text{AF}}_{\text{a}} }}{k\_I}} \right){\text{ + AF}}_{\text{b}} }} \\ \frac{{d({\text{AF}}_{\text{b}} )}}{dt} & = \frac{{k\_8 * {\text{IF}}_{\text{b}} *C_{\text{b}} }}{{k\_81{\text{ + IF}}_{\text{b}} }} + \frac{{k\_9*P_{\text{b}} * {\text{AF}}_{\text{b}} }}{{k\_91 + {\text{AF}}_{\text{b}} }} - k\_10 * {\text{AF}}_{\text{b}} \\ \end{aligned}$$

Appendix 2: list of model parameters

See the Table 2.

Table 2 Kinetic parameters referenced from [21]

Appendix 3: model variables and parameter values

See the Tables 3, 4, 5.

Table 3 Initial conditions

Table 4 Sets of values for erythropoiesis

Table 5 Values used for sensitivity analysis