Extended Data Fig. 5: Dimensionless analysis of the deterministic model for the mutual repressive system.
From: Unbalanced response to growth variations reshapes the cell fate decision landscape
(a) The model includes the global regulation of cell growth along with growth-dependent protein synthesis (αR and αG) and degradation (growth dilution λ under the assumption that protein turnover is negligible) as input stimuli and output of the system (Supplementary Note 1). The model for the mutual repressive system is typically written as \(\frac{{{{{\mathbf{d}}}}\left[ {{{\boldsymbol{R}}}} \right]}}{{{{{\mathbf{d}}}}{{{\boldsymbol{t}}}}}} = {{{\mathbf{\upalpha }}}}_{{{\boldsymbol{R}}}} \cdot {{{\boldsymbol{H}}}}_{{{\mathbf{R}}}}\left( {\left[ {{{\boldsymbol{G}}}} \right]} \right) - {{{\mathbf{\uplambda }}}} \cdot \left[ {{{\boldsymbol{R}}}} \right]\) and \(\frac{{{{{\mathbf{d}}}}\left[ {{{\boldsymbol{G}}}} \right]}}{{{{{\mathbf{d}}}}{{{\boldsymbol{t}}}}}} = {{{\mathbf{\upalpha }}}}_{{{\boldsymbol{G}}}} \cdot {{{\boldsymbol{H}}}}_{{{\mathbf{G}}}}\left( {\left[ {{{\boldsymbol{R}}}} \right]} \right) - {{{\mathbf{\uplambda }}}} \cdot \left[ {{{\boldsymbol{G}}}} \right]\), where \({{{\boldsymbol{H}}}}_{{{\boldsymbol{R}}}}\left( {\left[ {{{\boldsymbol{R}}}} \right]} \right)\) and \({{{\boldsymbol{H}}}}_{{{\boldsymbol{G}}}}\left( {\left[ {{{\boldsymbol{G}}}} \right]} \right)\) are the mutual repressive relations between TFs, which are described as two decreasing Hill functions of [R] (the reporter of TetR expression) and [G] (the reporter of LacI expression) with repression thresholds \({{{\boldsymbol{K}}}}_{{{{\boldsymbol{DR}}}},{{{\boldsymbol{DG}}}}}\), respectively. (b) By defining the expression capacity \({{{\tilde{\mathbf \upalpha }}}}_{{{{\boldsymbol{R}}}},{{{\boldsymbol{G}}}}}: = {{{\mathbf{\upalpha }}}}_{{{{\boldsymbol{R}}}},{{{\boldsymbol{G}}}}}/{{{\mathbf{\uplambda }}}}\), we would denote \(\left[ {{{{\tilde{\boldsymbol R}}}}} \right]: = \left[ {{{\boldsymbol{R}}}} \right]/{{{\tilde{\mathbf \upalpha }}}}_{{{\boldsymbol{R}}}}\), \(\left[ {{{{\tilde{\boldsymbol G}}}}} \right]: = \left[ {{{\boldsymbol{G}}}} \right]/{{{\tilde{\mathbf \upalpha }}}}_{{{\boldsymbol{G}}}}\) as the dimensionless concentrations, and g = λ·t as the generational time of the system, thereby obtaining: \(\frac{{d[{{{\tilde{\boldsymbol R}}}}]}}{{d{{{\boldsymbol{g}}}}}} = {{{\tilde{\boldsymbol H}}}}_{{{\boldsymbol{R}}}}([{{{\tilde{\boldsymbol G}}}}]) - [{{{\tilde{\boldsymbol R}}}}]\) and \(\frac{{{{{\mathrm{d}}}}[{{{\tilde{\boldsymbol G}}}}]}}{{d{{{\boldsymbol{g}}}}}} = {{{\tilde{\boldsymbol H}}}}_{{{\boldsymbol{G}}}}([{{{\tilde{\boldsymbol R}}}}]) - [{{{\tilde{\boldsymbol G}}}}]\). Due to the growth rate dependence, the dimensionless repressive thresholds \({{{\tilde{\boldsymbol K}}}}_{{{{\boldsymbol{DR}}}}}: = {{{\boldsymbol{K}}}}_{{{{\boldsymbol{DR}}}}}/{{{\tilde{\mathbf \upalpha }}}}_{{{\boldsymbol{R}}}}\) and \({{{\tilde{\boldsymbol K}}}}_{{{{\boldsymbol{DG}}}}}: = {{{\boldsymbol{K}}}}_{{{{\boldsymbol{DG}}}}}/{{{\tilde{\mathbf \upalpha }}}}_{{{\boldsymbol{G}}}}\) would effectively increase with the growth rates but with different magnitudes. Therefore, the dimensionless nullclines of the system where \({{{\mathbf{d}}}}\left[ {{{{\tilde{\boldsymbol R}}}}} \right]/d{{{\boldsymbol{g}}}} = 0\) and \({{{\mathbf{d}}}}\left[ {{{{\tilde{\boldsymbol G}}}}} \right]/d{{{\boldsymbol{g}}}} = 0\), such that \(\left[ {{{{\tilde{\boldsymbol R}}}}} \right] = {{{\tilde{\boldsymbol H}}}}_{{{\boldsymbol{R}}}}\left( {\left[ {{{{\tilde{\boldsymbol G}}}}} \right]} \right)\) and \(\left[ {{{{\tilde{\boldsymbol G}}}}} \right] = {{{\tilde{\boldsymbol H}}}}_{{{\boldsymbol{G}}}}\left( {\left[ {{{{\tilde{\boldsymbol R}}}}} \right]} \right)\) shifts as the growth rate changes, causing variations in the number of fixed points (Fig. 2b).
