Co-occurrence of resonant activation and noise-enhanced stability in a model of cancer growth in the presence of immune response

We investigate a stochastic version of a simple enzymatic reaction which follows the generic Michaelis-Menten kinetics. At sufficiently high concentrations of reacting species, that represent here populations of cells involved in cancerous proliferation and cytotoxic response of the immune system, the overall kinetics can be approximated by a one-dimensional overdamped Langevin equation. The modulating activity of the immune response is here modeled as a dichotomous random process of the relative rate of neoplastic cell destruction. We discuss physical aspects of environmental noises acting in such a system, pointing out the possibility of coexistence of dynamical regimes where noise-enhanced stability and resonant activation phenomena can be observed together. We explain the underlying mechanisms by analyzing the behavior of the variance of first passage times as a function of the noise intensity.

Figure 1

The Michaelis-Menten potential with parameters, $\beta =3,\theta =0.1,\Delta =0.02$. Labels: 엯, extremes of $U\left(x\right)$, $x=0,4,5$; ×, extremes of $U^{+}\left(x\right)$, $x=0,4.28,4.72$; ∗, extremes of $U^{-}\left(x\right)$, $x=0,3.83,5.17$.

Figure 2

MFPT as a function of correlation time $\tau$ of the dichotomous noise, for various starting points $x_{\mathrm{in}}$ in the Michaelis-Menten potential. The RA effect is well visible for initial positions near the metastable state and near the maximum of the potential. Initial positions, $x_{\mathrm{in}}=5$ (in the neighborhood of the right minimum), $x_{\mathrm{in}}=4$ (in the neighborhood of the maximum), $x_{\mathrm{in}}=3.5$ (left slope), $x_{\mathrm{in}}=3$ (left slope). The parameter values are $\beta =3$, $\theta =0.1$, $\Delta =0.02$, and $\sigma =0.1$.

Figure 3

MFPT as a function of the noise intensity $\sigma$. Influence of the multiplicative dichotomous noise on the NES effect for the same initial positions as in Fig. 2 and two values of correlation time of the dichotomous noise, namely, $\tau =0.01$ (solid line) and $\tau ={10}^8$ (dashed line). All the other parameters are the same as Fig. 2. The MFPTs in the absence of multiplicative noise (dotted lines), that is for an average fixed barrier, are compared with those calculated at very low correlation time $\tau$ (solid line).

Figure 4

MFPT for the higher (dotted line) and lower (short dashed line) position of the switching potential and their mean, as a function of the noise intensity $\sigma$. Although $\Delta$ is small, the NES effect in a static potential with $\beta +\Delta$ looks quite different from that obtained with $\beta -\Delta$. The mean MFPT of the static potentials (long dashed line) is compared with the MFPT calculated at very small switching frequency of the dichotomous noise. The initial position is $x_{\mathrm{in}}=3.5$, and all the other parameter values are the same as in Fig. 2.

Figure 5

NES vs RA: three-dimensional plot of MFPT as a function of noise intensity $\sigma$ of the additive noise and correlation time $\tau$ of the dichotomous noise. The initial positions of the Brownian particle and all the other parameter values are the same as in Fig. 2.

Figure 6

The coexistence of NES and RA phenomena in a three-dimensional plot of MFPT as a function of noise intensity $\sigma$ and correlation time $\tau$. The initial position of the Brownian particle is $x_{\mathrm{in}}=3.6$. Number of simulation runs $N={10}^4$. All the other parameter values are the same as in Fig. 2.

Figure 7

Cross sections of Fig. 6 for six values of the noise intensity $\sigma$, in the region of coexistence of RA and NES. The effect of resonant activation overlapping the noise-enhanced stability is well visible. The initial point is $x_{\mathrm{in}}=3.6$. Number of simulation runs $N={10}^4$. All the other parameter values are the same as in Fig. 2.

Figure 8

MFPT and its standard deviation, with error bars, as a function of the additive noise intensity $\sigma$ for two values of the correlation time $\tau$, namely (a) $\tau =1$ and (b) $\tau ={10}^5$. Number of simulation runs $N={10}^4$, initial position of the Brownian particle $x_{\mathrm{in}}=3.6$. All the other parameter values are the same as in Fig. 2. The asymmetry of appearance of error bars near the maximum in (a) is due to the large values of error bar in these points and to the log scale.