Adaptation Process over Several Environmental Conditions

<div><p>(A) Time series of protein expressions <i>x_i</i>(<i>t</i>) when the environmental condition is altered. The environmental conditions, i.e., substrates having nonzero <i>Y_i</i>, are changed at time points indicated by arrows.</p><p>(B) Change of growth rate <i>v_g</i> in the same time interval as (A). After the environmental changes, both expression levels of all proteins and the growth rate start to fluctuate until the cell finds a state of protein expression that realizes a high growth rate. In the simulation, the noise amplitude <i>σ</i> = 0.2.</p></div

The relationship between the noise amplitude and the growth rate obtained by the alternative model without the growth-dependence on EFR.

<p>The geometric mean of the growth rates attained by randomly generated regulatory networks is plotted as a function of the noise amplitude . The growth rates for different values of are superimposed by using different colors. In the intermediate range of the noise amplitude , the average growth rates obtained in the cases with are significantly higher than the case without EFR (), due to the selection of actively growing state by noise.</p

Selection of a Higher Growth State by Noise

<div><p>(A) Time series of protein expressions <i>x_i</i>(<i>t</i>). Ten out of 96 protein species are displayed. The vertical axis represents the expression levels of proteins, and the horizontal axis represents time.</p><p>(B) Change in growth rate <i>v_g</i> observed during the time interval shown in (A). Initially, the growth rate of the cell fluctuates due to the highly stochastic time course of protein expression. After a few short-lived nearly optimal states (c.f. 4,800 ∼ 5,600 time steps), the cell finds a state of protein expression that realizes a high rate of growth. The parameters are <i>θ</i> = 0.5, <i>μ</i> = 10, <i>ρ_a</i> = <i>ρ_i</i> = 0.03, <i>ɛ</i> = 0.1, and <i>D</i> = 0.1. In addition, we enhanced the rate of positive autoregulatory paths, i.e., <i>W_ii</i> = 1 for <i>i</i>-th gene, so that the regulatory network has multiple attractors. In the simulations, 30% of activating paths are chosen as autoregulatory paths.</p></div

Multicellular model of T cell response.

<p>(a) A schematic illustration of the model. Each APC presents a self or non-self antigen. T cells are associated with APC in a stochastic manner, while the dissociation rate <i>k</i>_off depends on the combination of antigen and TCR expressed on the T cell. These T cells divide only when they are associated with APC, which is suppressed by Treg cells in the environment. Tconv and Treg cells are supplied from outside the system constantly, in a ratio of 9:1, which is based on experimental observation [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0163134#pone.0163134.ref010" target="_blank">10</a>]. Simultaneously, T cells which are not attached to APCs are discarded in a constant rate. (b) The distribution of <i>k</i>_off for Tconv cells. Two distributions of and of Tconv cells supplied to the system are plotted. The parameters are <i>μ</i>^Tconv,self = −3, <i>μ</i>^{Tconv,non−self} = −3.75 (i.e., Δ^Tconv = 0.75), and <i>σ</i> = 1. (c) The distribution of <i>k</i>_off for Treg cells supplied to the system. The parameters are <i>μ</i>^Treg,self = −3.75, <i>μ</i>^{Tconv,non−self} = −3 (i.e., Δ^Treg = 0.75), and <i>σ</i> = 1.</p

An example of adaptation process with EFR.

<p>(a) Time series of expression levels . Eight of the 40 gene expression levels are displayed.(b) Time series of the epigenetic factors that correspond to the expression levels displayed in (a). (c) Change in the growth rate . (d) (inset of (c))The relationship between the noise amplitude and the growth rate . The geometric mean of the growth rates attained by randomly generated regulatory networks and initial conditions is plotted as a function of the noise amplitude . The error bars represent the geometric standard deviation. The blue dotted line indicates the growth rate in the case of a random selection of the state, where the expression level takes the values 0 and 1 with equal probability. The growth rate was significantly higher than the random expression pattern, for example when (the number of data was 10000; determined by U-test). The parameter values used here are , , , , , , , and . The target expression patterns used in the growth rate calculation were determined randomly. Unless otherwise mentioned, these values were used throughout all the figures. The presented results were independent of specific model parameters, and were observed in a wide range of parameter values. We selected above parameter values to present general features of this model.</p

Robust and Accurate Discrimination of Self/Non-Self Antigen Presentations by Regulatory T Cell Suppression

<div><p>The immune response by T cells usually discriminates self and non-self antigens, even though the negative selection of self-reactive T cells is imperfect and a certain fraction of T cells can respond to self-antigens. In this study, we construct a simple mathematical model of T cell populations to analyze how such self/non-self discrimination is possible. The results demonstrate that the control of the immune response by regulatory T cells enables a robust and accurate discrimination of self and non-self antigens, even when there is a significant overlap between the affinity distribution of T cells to self and non-self antigens. Here, the number of regulatory T cells in the system acts as a global variable controlling the T cell population dynamics. The present study provides a basis for the development of a quantitative theory for self and non-self discrimination in the immune system and a possible strategy for its experimental verification.</p></div

The Relationship between the Noise Amplitude <i>σ</i> and the Growth Rate <i>v_g</i>

<p>Starting from randomly chosen initial conditions against the noise amplitude <i>σ</i> ranging 10⁻⁴ < <i>σ <</i>3, the growth rates <i>v_g</i> after 10⁵ time steps are plotted. In the intermediate range <i>σ</i> of the noise strength 10⁻² < <i>σ <</i>1, cellular states with high growth rates are selected among a huge number of possible cellular states, as depicted in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.0040003#pcbi-0040003-g002" target="_blank">Figure 2</a>.</p

Evolution of the highest fitness as a function of the generation.

<p>100 independent evolutionary simulations were performed, and the average of the highest fitness value in each generation is plotted along with the standard deviation. The fitness of a regulatory network is defined as the geometric average of the growth rate, , in different environments. For the cases with EFR, the noise amplitude, , was set to 0 and 1, respectively. For each generation, the number of parent networks was set to , while for each parent network mutant networks were generated by randomly replacing a regulatory path. The total number of regulatory paths was fixed to 400. In the cases without EFR, the epigenetic factor was fixed to 0.</p

The distributions of growth rates in a novel environment for evolved networks and random networks.

<p>100 evolved networks obtained after 100 generations with EFR (; see Fig. 2) and 100 random networks were used for the calculations. For each network, the growth rates were calculated using 50 randomly chosen target expression patterns that were not used for the evolutionary simulations. For all simulations in this figure, is set to 1 and EFR is incorporated in the expression dynamics. The difference in the growth rates was statistically significant (; determined by U-test).</p

Schematic Representation of Our Cell Model

<p>The model consists of two networks, i.e., a gene regulatory network and a metabolic network. As a schematic example, simple networks consisted of <i>n</i> = 7 genes and <i>m</i> = 6 metabolic substrates are shown. The red arrows in the regulatory network represent activation of expressions, while green lines with blunt ends represent inhibition. The arrows from a gene to itself mean autoregulation of expressions. As a result of these regulatory interactions, the dynamics of expression levels of proteins have multiple attractors. The metabolic reactions, represented by blue arrows, are controlled by expression levels of corresponding proteins. The correspondence between metabolic reactions and gene products (proteins) are shown by the thin black arrows. The regulatory matrix <i>W_ij</i> of the presented network takes <i>W</i>₂₁ = <i>W</i>₃₂ = <i>W</i>₃₃ = <i>W</i>₄₅ = <i>W</i>₅₆ = <i>W</i>₆₇ = <i>W</i>₇₇ = 1, <i>W</i>₂₄ = <i>W</i>₅₃ = <i>W</i>₅₇ = −1, and 0 otherwise. The reaction matrix <i>Con</i>(<i>i</i>,<i>j</i>,<i>k</i>) metabolic network takes a value 1 for the elements (1,3,1)(2,3,2)(3,4,3)(6,3,4)(4,5,5)(6,4,6)(5,6,7), and 0 otherwise. The choice of <i>n</i> = 7 in the figure is only for schematic illustration, and in the actual simulation we used much larger networks with <i>n</i> = 20 ∼ 100. In the present paper, we adopt a much larger network with <i>n</i> = 96 genes and <i>m</i> = 32 substrates.</p