Biology core problems

Problem 1 – Resonant cell growth#

A culture of cells divides according to a resonance-driven growth law

$$ G(t) = G_0 e^{D_3 τ_r t} $$

where $$G_0$$ is the initial cell count, $$D_3$$ is a triadic growth operator, and $$τ_r$$ is a resonant-time parameter reflecting environmental rhythm.

The biologist finds that increasing $$τ_r$$ by 10% yields faster growth. If $$τ_r$$ is changed from $$τ_r$$ to $$1.1τ_r$$ by what factor does the growth factor $$e^{D_3 τ_r t}$$ change at a fixed time $$t$$?

Problem 2 – Protein folding stability under environmental resonance#

Protein stability in a certain experiment is modeled as

$$ P = \frac{ΛΘ}{D_9} $$

where $$ΛΘ$$ encodes environmental temperature/stress coupling and $$D_9$$ encodes a triadic destabilizing factor.

During an environmental shift, $$D_9$$ increases due to added noise. To keep protein stability $$P$$ constant, how must $$Θ$$ change in relation to the change in $$D_9$$, assuming $$Λ$$ stays fixed?

Problem 3 – Neural oscillation coupling#

Neurons in a small brain region fire in triadic bursts with resonance frequency

$$ f_n = T_f D_6 $$

where $$D_6$$ is a structural triad linked to network connectivity, and $$T_f$$ is a frequency elevation factor modulated by neuromodulators.

If the brain region enters a high-attention state that requires $$f_n$$ to increase by 15%, and $$D_6$$ is unchanged, by what factor must $$T_f$$ be adjusted?