Table 1 List of mathematical symbols and notations.
From: Theoretical study of the impact of adaptation on cell-fate heterogeneity and fractional killing
Symbol | Description | Equations/Figures |
|---|---|---|
\(x_i^l\), \(\vec {x}^l\) | Concentration of biochemical species i of cell l | Eq. (1) |
\(k_i^l\), \(\vec {k}^l\) | Biochemical network parameter i of cell l | Eq. (1) |
\(a_j\), \(\nu _{ji}\) | Rate and stoichiometries of the biochemical reaction j | Eq. (1) |
\(\xi _j^l\) | Langevin noise associated to reaction j in cell l | Eq. (1) |
\(\sigma\) | Standard deviation of random variable | Eq. (1) |
\(s^l\) | Stimulus (e.g., stress) level of cell l | Eq. (1) |
\(s_{sn}\) | Stimulus level associated with saddle-node bifurcation | Fig. 2 |
\(P(\vec {x},t)\) | Time-dependent probability distribution function in state space | Eq. (2) |
\(P_{D/Death}\) | Decision (e.g., death) probability | Eq. (2) |
\(s_{50}\) | Stimulus level inducing \(50\%\) of fate probability | Eq. (3) |
\(t^*\) | Measurement time for \(P_D\) | Eq. (2) |
\(\eta\) | Fractionality index | Eq. (3) |
\(x_{1,2,3}\) | Adaptive (e.g., damage/repair) and fate-decision (e.g., death) species | Eq. (4) |
\(\beta ,\tau\) | Adaptation strength and timescale | Eq. (4) |
\(\vec {x}_{st1,2/sn/sad}\) | Stable/saddle-node/saddle fixed point associated with bistability | |
\(\mathcal{W}^{s/u}(\vec {x})\) | Stable/unstable manifold of the fixed point \(\vec {x}\) | |
\(s_{c}\) | Critical stimulus level without noise \(s_c=s_{50}(\sigma =0)\) | Fig. 2 |
\(\vec {x}_c(t,s_c)\) | Critical trajectory | |
\(\vec {y}(t)\), \(y_N\) | Small deviations of \(\vec {x}(t)\) from \(\vec {x}_c(t)\) | |
\(\Pi (t,t')\) | Principal fundamental matrix | Eq. (7) |
\(U(x),\Delta ,r_K\) | Effective potential, barrier height and Kramers rate |
