Fig 1.
Energetic description of interacting cells stocks embedded in an organism.
The diagram shows competing cells stocks as determined by the main process of ATP production, biosynthesis, and by the antagonistic interaction. It shows the associated flows of dissipated and useful energy, the latter being able to generate feedback for self-regulation on a higher hierarchical scale. The flow of energy and resources directed toward the rest of the organism is reduced by the inflows of the sub-system of competing cells stocks. The picture represents the interaction of plasma cells at myelomas onset which we define as being the core of the representation of Multiple Myeloma in the energy system language.
Fig 2.
Self-regulating cell population system.
The diagram in (A) represents a self-regulating cell population seen as a system that produces and consumes ATP from primary energy and microenvironmental resources, respectively through the processes of ATP production and biosynthesis. The cells stock exerts feedback on the ATP production process needed for proliferation. From the biosynthesis process develops feedback on growth represented by the population’s ability to: 1) regulate limiting factors coming from the interaction with the micro-environment, as cofactors and dissolved ions; 2) produce functional proteins, molecular machines and enzymes that enable reactions and control their rates. From (A) we define in (B) the modeled quantities: the primary energy inflow, Jin; the ATP production efficiencies, η; the cells stock Q that grows according to with P(Q) = ηJin = rQ·(1−Q/K), net power inflow (with r intrinsic growth rate, K carrying capacity), and R(Q) = Q/τ energy outflow, being τ the proteome turnover time; the stock of resources, N = K−Q, for the mass balance to hold; the total heat flow Jh = P/η = Jin in steady state.
Fig 3.
Stable stationary states for normal and neoplastic plasma cells.
Panel (A) shows the actual steady states (colored dots) occupied by the PCs system in the state space defined by the bell-shaped curves P(Q) = r·Q·(1−Q/K) and Jh(Q) = P(Q)/η, depending on the intrinsic growth rates r of normal PCs (r = 1.39·10−2h−1, in blue) and malignant PCs (r = 1.43·10−2h−1−1.53·10−2h−1, red hues), with K = 4.05·1021 ATPeq, τ = 72 h (parameter estimation in Methods). Panel (B) shows the associated stability curve V(Q) defined by dV(Q)/dQ = −dQ/dt (Eq 8, derived in Methods).
Fig 4.
Competing cell populations system.
The diagram in (A) represents a system made of normal cells competing with its malignant (cancerous) counterpart for common limited resources. Each single population is seen as a system that produces and consumes ATP, following the model in Fig 2. In addition, populations compete for space in the tissue and interact by means of an effective biochemical interaction which favors the death of the competitor. Each process dissipates energy under the form of heat that sinks at the bottom of the diagrams. Panel (B) shows the mathematical model for the system. It highlights the physical observables: external primary energy inflow, Jin, the net power inflows, P1 and P2, for the respective cells’ stocks, Q1 (normal PCs) and Q2 (neoplastic PCs), and the heat flow, Jh, the stock N of resources (shared by Q1 and Q2); the biochemical antagonistic interaction is parametrized by I1 and I2.
Fig 5.
Coexistence and regime shifts for interacting normal and neoplastic plasma-cells.
Panel (A) shows the time evolution of stock Q1 (in blue) and Q2 (in red hues) according to Eqs 3 and 4 on yearly time scale for configurations defined by the different values of r2 = 1.43·10−2−1.53·10−2h−1 for neoplastic PCs as in Fig 4, and with K = 4.05·1021 ATPeq, r1 = 1.39·10−2h−1, τ1 = τ2 = 72 h, α1 = α2 = 0, Q1(0) = Qss = 0.02·K, Q2(0) = 5·1010 ATPeq. Panel (B) shows the time evolution of stock Q1 (blue) and Q2 (red) for configurations defined by the different values of α1 = c·r1/K and α2 = c·r2/K with c = 0, 1, 2 (solid, dashed, dot dashed lines) and by r2 = 1.47·10−2h−1, r1 = 1.39·10−2h−1, K = 4.05·1021 ATPeq, τ1 = τ2 = 72 h, Q1(0) = Qss = 0.02·K, Q2(0) = 5·1010 ATPeq.
Fig 6.
Diagram and model for autocatalytic growth.
This figure shows how to derive differential equations for the evolution of the stock Q from the elementary diagram in A. Diagram (A) represents a system with self-reinforcing feedback. Panel (B) shows the dynamical model associated with the diagram in (A). The time evolution of the stock is modeled as a first-order time dependent differential equation for Q, that defines the system state at any time.
Fig 7.
Self-limiting autocatalytic cycle model.
The figure shows how to constrain and balance the stock-flow model used to describe the growth of a single cell population. Diagram (A) represents a self-limiting autocatalytic cycle with the relevant flows interacting through the production process and proportional to E·N·Q and the outflow proportional to Q. Upper right a summary of the system dynamics for the stock N and Q with the first law constraining the kinetic coefficients. Panel (B) shows the non-equilibrium thermodynamics representation for the self-limiting cycle in diagram (A). We define: the primary energy inflow, Jin = k0ENQ; the stock of limiting factors, N, with its inflow and outflow from the external environment JN,in = JN,out; the production and consumption processes efficiencies, η and ξ; the cells stock, Q, with its net power inflow, P = k1ENQ = ηJin, and energy outflow, R = k2Q = Q/τ; the recycling and control feedback flows, ξ·R and P–(1–ξ)·R; the total heat flow, Jh = P/η = Jin, which is the sum of the heat flows associated to the production, P/η–(1–ξ)·R, and consumption process, (1–ξ)·R. Jh is a measure of the irreversibility of the modeled growth process.
Table 1.
Relevant quantities and parametrization for the growth model of a single cell population.
Fig 8.
Stable stationary states and dynamics of the self-limiting autocatalytic cycle.
Panel (A) shows the model state space defined by the curves for the power (bold black), P(Q) = r·Q·(1−Q/K), the heat flow (black), Jh(Q) = P(Q)/η and the stability potential (dot dashed), V(Q) = a·Q3+b·Q2 (coefficients in Eq 8). The colored dots on the curves represent the steady-states for P, Jh and V as a function of Q (0<Q<K) for the three possible different growth regimes, respectively, defined by the product r·τ: sub-optimal, r·τ = 1.5 (azure), optimal or maximum power state (Pmax), r·τ = 2 (blue), and super-optimal, r·τ = 10 (dark-blue). This is associated with the stock N = K-Q (dashed lines) that decreases for increasing Q to fulfill the mass balance constraint. Panels (B) and (C) show the time evolution associated with the configurations defined by the three different growth regimes (azure, blue, dark blue), respectively, for the stocks Q and N (dashed), and for the flows P and Jh. Panel (D) shows the application of the formulation to estimate the stationary states of normal and neoplastic plasma cells.
Fig 9.
Competition dynamics and energetic constraints.
This figure shows how to constrain and balance the stock-flow model used to describe the growth of interacting cell populations. Both (A) and (B) represent a system of stocks competing for space. In diagram (A), the upper right shows the mass balance constraint on N, with power inflows proportional to ENQ1 or Q2 and respectively with outflows proportional to Q1 or Q2. The interaction term is a Lotka-Volterra-like proportional to Q1·Q2. The primary energy inflow, Jin = k0,1ENQ1+k0,2ENQ2 is transformed into useful power flows P1 = k1ENQ1 = η1Jin,1 and P2 = k2ENQ2 = η2Jin,2 for the two stocks with efficiencies, η1 and η2; The efficiencies ξ1 and ξ2 control the partitioning energy outflows R1 = k3Q1 = Q1/τ1; and R2 = k4Q2 = Q2/τ2 between recycling feedback flows and the heat flows. The total heat flow Jh = Jin = P1/η1+P2/η2 sinks out of the boundary at the bottom of the diagram, with 0<δ<1 being the phenomenological efficiency for the direct interaction term.
Table 2.
Biophysical quantities and parameterization for competing cell populations.
Table 3.
Parameter setting.
Fig 10.
Simulation of the power flows and the total heat flow for competing cells stocks.
It shows the time evolution on the yearly time scale for the power flows P1 (blue) and P2 (red), and the total heat flow Jh (black), for the different values of α1/α2 represented by the different line styles, as in the legend. The setting for the other parameters follows the values reported in Table 3: K = 4.05·1021 ATPeq, Q1,0 = 8.1·1019 ATPeq, Q2,0 = 5·1010 ATPeq, η = 40%, τ1 = τ2 = 72 h, r1 = 1.39·10−2h−1, r2 = 1.53·10−2h−1.