🧬 Biology — TFT_3Pack Example Suite

Problem 1 — Resonant Cell Growth

G(t) = G₀ e^{D₃ τ_r t}

If \( τ_r \) increases by 10%, how does the growth factor change at fixed time \( t \)?

Problem 2 — Protein Folding Stability

P = ΛΘ / D₉

If \( D₉ \) increases, how must \( Θ \) change to keep \( P \) constant?

Problem 3 — Neural Oscillation Coupling

fₙ = T_f D₆

If \( fₙ \) must increase by 15%, how must \( T_f \) change?

Solution 1 — Cell Growth

Growth factor multiplies by \( e^{0.1 D₃ τ_r t} \).

Solution 2 — Protein Stability

\( Θ' = Θ \cdot (D₉' / D₉) \).

Solution 3 — Neural Frequency

\( T_f' = 1.15 T_f \).

Problem 4 — Species Competition

A(t) = A₀ e^{D₃ τ_r t}
B(t) = B₀ e^{D₆ τ_r t}
        

Find the crossover time \( t^* \) and describe how it shifts with τ_r.

Problem 5 — Enzyme Activity

E_act = X e^{-1/(ΛΘ)}

How does doubling Λ affect the exponent and activity?

Problem 6 — Circadian Rhythm

ω_eff = T_f / τ_r

Find T_f needed for a 24-hour cycle.

Problem 7 — Logistic Growth Midpoint

N(t) = K / (1 + e^{-D₃(t - τ_r)})

Evaluate N(τ_r) and describe how τ_r shifts the midpoint.

Problem 8 — Signal Cascade

S₁ = F₃
S₂ = D₃(τ_r) S₁
S₃ = T_f S₂
        

Analyze how changes in τ_r and T_f affect S_out.

Diagram 1 — Cell Growth Pipeline

G₀ → × e^{D₃ τ_r t} → G(t)
        

Diagram 2 — Protein Stability Loop

Λ + Θ → ΛΘ → ÷D₉ → P
        

Diagram 3 — Neural Oscillation

T_f × D₆ → fₙ