Table 1 Combined description of variables, symbols, and GBM scRNA-seq datasets used in the study.

BT_S1

Neoplastic cells from tumor core of patient BT_S1

265

BT_S2

Neoplastic cells from tumor core of patient BT_S2

502

BT_S4

Neoplastic cells from tumor core of patient BT_S4

134

BT_S6

Neoplastic cells from tumor core of patient BT_S6

126

BT_All

All patients neoplastic cells from tumor core

1027

BT_Regular

All patients regular cells from tumor core and periphery

2489

k

Self-degradation constant

\(k = 1\)

S

Hill function inflection point value

\(0.5 \le S \le 4\), in steps of 0.5

n

Hill function transition intensity

\(1 \le n \le 4\), in steps of 0.5

\(a_i\)

Activation coefficients for gene i

\(0.01 \le a_i \le\)

10

\(b_i\)

Inhibition coefficients for gene i

\(0.01 \le b_i \le\)

10

\(\eta\)

Proportionality constant in multiplicative noise

\(\eta = 0.001\)

\(t_{sim}\)

Simulation time

\(t_{sim} = 200\) a.u.

\(\Delta t\)

Time steps used in simulations

\(\Delta t = 0.05\) a.u.

\(N_{method}\)

Numerical method used for solving SDEs

Euler-Maruyama

\(\textbf{X}_\alpha\)

Centroid vector for cluster \(\alpha\)

X_α = {X_α,1, X_α,2, ..., X_α,n }

\(\mathbf {X_{sim}}(0)\)

Initial condition for simulations, using centroids and/or noise

X_sim(0)=

\({\textbf{X}}_\alpha + \beta \cdot \mathbf {\varepsilon ^0}\) or \(\mathbf {X_{sim}}(0) = \mathbf {\varepsilon ^1}\)

\(\mathbf {\varepsilon ^0}\)

Gaussian noise applied to initial conditions

\(\mathbf {\varepsilon _{i}^{0}} \sim \mathscr {N}\)

(0.5, 0.1)

\(\mathbf {\varepsilon ^1}\)

Uniformly distributed random noise for initial conditions

\(\mathbf {\varepsilon _{i}^{1}} \sim U(0, 10)\)

\(\beta\)

Proportionality constant for perturbation in initial conditions

0 and 1