Entropy, Information & Aging

Derivation: Shannon Entropy and the Information Content of DNA

Shannon entropy for a discrete random variable with $n$ possible outcomes, each with probability $p_i$, is:

$$H = -\sum_{i=1}^{n} p_i \log_2(p_i) \quad \text{(bits)}$$

For DNA, there are 4 possible bases at each position (A, T, G, C). If all bases are equally probable ($p_i = 1/4$):

$$H_{\max} = -4 \times \frac{1}{4}\log_2\frac{1}{4} = -\log_2\frac{1}{4} = \log_2 4 = 2 \text{ bits per base pair}$$

The human genome contains approximately $3.2 \times 10^9$ base pairs, so the maximum information content is:

$$I_{\max} = 3.2 \times 10^9 \times 2 = 6.4 \times 10^9 \text{ bits} \approx 6.4 \text{ Gbits} \approx 800 \text{ MB}$$

However, the actual information content is less because: (1) base frequencies are not uniform (GC content varies), (2) there are extensive repetitive sequences (~50% of the genome), and (3) only ~1.5% codes for proteins. The effective information content is estimated at roughly 50–100 MB.

Mutual information between gene expression and environment quantifies how much information about the environment is encoded in gene expression patterns:

$$I(X;Y) = \sum_{x,y} p(x,y) \log_2 \frac{p(x,y)}{p(x)p(y)}$$

where $X$ represents gene expression states and $Y$ represents environmental conditions. This equals $I(X;Y) = H(X) + H(Y) - H(X,Y)$. For a perfectly adapted organism, the mutual information between its internal state and the environment is maximized — the organism is a model of its environment, as originally proposed by Conant and Ashby (1970).

Derivation: Entropy Production Rate and Aging

According to non-equilibrium thermodynamics, the entropy production rate of a living system can be written as a sum of thermodynamic forces $X_k$ and fluxes $J_k$:

$$\sigma = \frac{dS_i}{dt} = \sum_k J_k X_k \geq 0$$

For metabolic processes, the dominant contribution comes from the dissipation of chemical free energy:

$$\sigma \approx \frac{1}{T}\sum_r \dot{\xi}_r (-\Delta G_r)$$

where $\dot{\xi}_r$ is the rate of reaction $r$ and $\Delta G_r$ is its free energy change. Prigogine's minimum entropy production theorem states that for a system near equilibrium with fixed boundary conditions, the steady state minimizes the entropy production rate:

$$\frac{d\sigma}{dt} \leq 0 \quad \text{(near equilibrium, linear regime)}$$

In the context of aging, we can model the organism's state as characterized by a set of order parameters $\{q_i\}$ that describe its organization. The free energy of maintaining this organization is:

$$G_{\text{maint}} = \sum_i G_i(q_i)$$

Aging corresponds to the gradual increase in the cost of maintaining organization, as repair mechanisms become less efficient. The entropy production rate increases with age as the system departs from its optimal steady state:

$$\sigma(t) = \sigma_0 + \alpha t + \beta t^2$$

where $\sigma_0$ is the basal entropy production, $\alpha$ represents linear degradation, and $\beta$ represents accelerating deterioration. Disease states correspond to sudden increases in $\sigma$ — departure from the minimum entropy production steady state.

The total entropy produced over a lifetime is $S_{\text{total}} = \int_0^{t_{\text{life}}} \sigma(t) \, dt$. Interestingly, the lifetime entropy production per unit mass is approximately constant across mammalian species at about $\sim 10^4$ kJ/(kg·K), consistent with a universal "entropy budget" for life.

Derivation: Mitochondrial ROS Production and Damage Accumulation

The Nernst equation for a mitochondrial redox couple determines the tendency for electron leak:

$$E = E^{\circ\prime} - \frac{RT}{nF}\ln\frac{[\text{Red}]}{[\text{Ox}]}$$

When the electron transport chain is highly reduced (high NADH/NAD$^+$ ratio), single electrons can "leak" to molecular oxygen:

$$\text{O}_2 + e^- \to \text{O}_2^{\bullet-} \quad (E^{\circ\prime} = -0.33 \text{ V})$$

The rate of superoxide production depends on the redox state of the electron carriers. For a simplified model where electron leak rate is proportional to the fraction of reduced carriers:

$$R_{\text{ROS}} = k_{\text{leak}} \cdot f_{\text{red}} \cdot [\text{O}_2]$$

where $f_{\text{red}}$ is the fraction of reduced ubiquinone. Under normal conditions, about 0.1–2% of electrons leak to form superoxide. The steady-state ROS concentration is determined by the balance of production and scavenging:

$$\frac{d[\text{ROS}]}{dt} = R_{\text{ROS}} - k_{\text{SOD}}[\text{SOD}][\text{O}_2^{\bullet-}] - k_{\text{GPx}}[\text{GPx}][\text{H}_2\text{O}_2] - k_{\text{cat}}[\text{Cat}][\text{H}_2\text{O}_2]$$

At steady state ($d[\text{ROS}]/dt = 0$), the ROS concentration is:

$$[\text{ROS}]_{ss} = \frac{R_{\text{ROS}}}{k_{\text{scav}}[\text{Antioxidants}]}$$

The damage accumulation model describes oxidative damage as the integral of net damage rate over time:

$$D(t) = D_0 + \int_0^t (k_{\text{damage}} \cdot [\text{ROS}]_{ss} - k_{\text{repair}}) \, d\tau$$

If repair capacity declines with accumulated damage ($k_{\text{repair}}(D) = k_{r,0}(1 - D/D_{\max})$):

$$\frac{dD}{dt} = k_d \cdot [\text{ROS}]_{ss} - k_{r,0}\left(1 - \frac{D}{D_{\max}}\right)$$

This nonlinear ODE produces the characteristic sigmoidal aging curve: slow damage accumulation when young (repair keeps up), followed by accelerating damage when repair capacity is overwhelmed. Death occurs when $D \to D_{\max}$. The mitochondrial "vicious cycle" hypothesis adds that damaged mitochondria produce more ROS, creating positive feedback: $R_{\text{ROS}}(D) = R_0(1 + \gamma D)$.

Derivation: Hayflick Limit from Telomere Shortening

Telomeres are repetitive DNA sequences (TTAGGG)$_n$ at chromosome ends that shorten with each cell division due to the end-replication problem. After $n$ divisions:

$$L(n) = L_0 - n \cdot \Delta L$$

where $L_0$ is the initial telomere length (typically 10–15 kbp in humans), and $\Delta L \approx 50$–200 bp per division (varies by cell type). Cells enter senescence when telomeres reach a critical length $L_{\text{crit}} \approx 4$–6 kbp, triggering the DNA damage response.

The Hayflick limit is therefore:

$$n_{\max} = \frac{L_0 - L_{\text{crit}}}{\Delta L}$$

For typical values $L_0 = 12$ kbp, $L_{\text{crit}} = 5$ kbp,$\Delta L = 100$ bp:

$$n_{\max} = \frac{12000 - 5000}{100} = 70 \text{ divisions}$$

Telomerase kinetics: In stem cells and cancer cells, telomerase extends telomeres. The dynamics become:

$$\frac{dL}{dt} = k_{\text{tel}} - \frac{\Delta L}{\tau_{\text{div}}}$$

where $k_{\text{tel}}$ is the telomerase elongation rate and $\tau_{\text{div}}$is the cell division time. The immortalization condition is:

$$k_{\text{tel}} \geq \frac{\Delta L}{\tau_{\text{div}}}$$

When this condition is satisfied, telomere length is maintained indefinitely. The steady-state telomere length with telomerase is:

$$L_{ss} = L_{\text{crit}} + \frac{(k_{\text{tel}} \cdot \tau_{\text{div}} - \Delta L) \cdot \tau_{\text{div}}}{\Delta L} \cdot L_{\text{crit}}$$

More precisely, if we model telomerase as adding $\delta$ bp per division when active (with probability $p_{\text{tel}}$): $\langle\Delta L_{\text{net}}\rangle = p_{\text{tel}} \cdot \delta - \Delta L$. The Hayflick limit is abolished when $p_{\text{tel}} \cdot \delta > \Delta L$. In cancer, telomerase reactivation (85% of cancers) or the ALT pathway (15%) achieves this, enabling unlimited replication.

Derivation: MEPP and Biological Organization

Consider a system receiving energy flux $\Phi_{\text{in}}$ from the environment. The entropy production rate depends on the internal organization:

$$\sigma = \frac{\Phi_{\text{in}}}{T_{\text{hot}}} - \frac{\Phi_{\text{out}}}{T_{\text{cold}}} + \frac{dS_{\text{internal}}}{dt}$$

At steady state ($\Phi_{\text{in}} = \Phi_{\text{out}} = \Phi$):

$$\sigma = \Phi\left(\frac{1}{T_{\text{cold}}} - \frac{1}{T_{\text{hot}}}\right)$$

Maximizing $\sigma$ means maximizing the energy flux $\Phi$ through the system. A more complex internal structure (with more degradation pathways) can process more energy flux, thus producing more entropy. This connects to Jaynes' maximum entropy inference: the most probable macroscopic state is the one consistent with constraints that maximizes entropy.

For a biological system with internal degrees of freedom $\{q_i\}$ subject to constraints $\{f_k(q_i) = c_k\}$, the MEPP predicts that the system evolves to maximize:

$$\mathcal{L} = \sigma(\{q_i\}) - \sum_k \lambda_k (f_k(\{q_i\}) - c_k)$$

where $\lambda_k$ are Lagrange multipliers for the constraints. This leads to thePhysiological Fitness Landscape: the fitness of an organism can be mapped as a function in the space of physiological parameters, with the optimum corresponding to the maximum entropy production state consistent with genetic and environmental constraints.

The MEPP has been applied to explain: (1) the evolution of metabolic networks that maximize energy dissipation, (2) the emergence of multicellularity as a strategy for increased entropy production, (3) the scaling of metabolic rate with body size (Kleiber's law as an optimization result), and (4) ecosystem structure and succession. While controversial, the MEPP provides a unifying thermodynamic framework for understanding biological organization and its inevitable degradation in aging.

Derivation: Information Loss Through Mutation Accumulation

Somatic mutations accumulate with age at a rate of approximately $\mu \approx 40$mutations per cell division per genome. The total mutation burden after $n$divisions is:

$$M(n) = \mu \cdot n$$

The information loss per mutation depends on the context. For a coding sequence, each point mutation changes one codon, potentially altering the amino acid. The information change per mutation in a random sequence position is:

$$\Delta I = -\log_2\left(\frac{1}{3}\right) \approx 1.58 \text{ bits (choosing one of 3 alternative bases)}$$

However, most of the genome is non-coding, and many coding mutations are synonymous (silent) or occur in non-essential genes. The effective information loss rate depends on the fraction of the genome under selective constraint. For the ~5% of the human genome under purifying selection:

$$\dot{I}_{\text{loss}} = 0.05 \times \mu \times 1.58 \text{ bits per division}$$

Over a lifetime of approximately $10^{16}$ total cell divisions across all tissues, with most cells dividing $\sim$40 times, individual cells accumulate$\sim$1600 somatic mutations. Most are passengers; the probability of a driver mutation in a specific oncogene is:

$$P(\text{driver}) = 1 - (1 - p_{\text{driver}})^{M(n)}$$

where $p_{\text{driver}} \sim 10^{-7}$ per mutation. This creates a direct link between aging (mutation accumulation) and cancer risk, which increases approximately as the 5th–6th power of age for most cancers, consistent with the multi-hit model requiring 5–6 driver mutations.

Gompertz Law of Mortality and Thermodynamic Interpretation

Benjamin Gompertz (1825) observed that the human mortality rate increases exponentially with age after maturity:

$$h(t) = h_0 \cdot e^{\gamma t}$$

where $h(t)$ is the hazard function (instantaneous mortality rate), $h_0$is the baseline mortality, and $\gamma \approx 0.085$ per year for humans (mortality doubles every $\ln 2 / 0.085 \approx 8.2$ years). The survival function is:

$$S(t) = \exp\left(-\frac{h_0}{\gamma}(e^{\gamma t} - 1)\right)$$

The thermodynamic interpretation connects $\gamma$ to the rate of entropy accumulation: as damage increases exponentially, the system's ability to maintain homeostasis degrades exponentially, leading to exponentially increasing failure probability. The Gompertz constant $\gamma$ correlates with mass-specific metabolic rate across species, supporting the connection between metabolic entropy production and aging rate.