Systems Biology & Frontiers

Question 1

(a) (4 marks)

• $\beta \frac{x^n}{K^n+x^n}$: saturating production driven by autoactivation; the fraction is the probability the promoter is bound/active (1 mark).
• $\gamma x$: first-order degradation/dilution, rate proportional to concentration (1 mark).
• $n>1$ indicates cooperative binding of multiple TF molecules (or multimerization) giving a sigmoidal/switch-like response rather than hyperbolic (1 mark).
• Emergence: the sharp switch/bistability is a system-level property not present in any single molecule — arising from the nonlinear feedback loop, illustrating holistic systems behavior (1 mark).

(b) (6 marks) Set $\frac{dx}{dt}=0$ with $n=2,\beta=4,K=1,\gamma=1$: $4\frac{x^2}{1+x^2} = x$ (1) $x=0$ is a root (1). For $x\neq0$ divide by $x$: $\frac{4x}{1+x^2}=1 \Rightarrow 4x = 1+x^2 \Rightarrow x^2-4x+1=0$ (2) $x = \frac{4\pm\sqrt{16-4}}{2} = 2\pm\sqrt{3}$ (1) Steady states: $x^*_0=0,\; x^*_-=2-\sqrt3\approx0.268,\; x^*_+=2+\sqrt3\approx3.732$ (1).

(c) (6 marks) Let $f(x)=4\frac{x^2}{1+x^2}-x$. Then $f'(x) = 4\cdot\frac{2x(1+x^2)-x^2\cdot2x}{(1+x^2)^2}-1 = \frac{8x}{(1+x^2)^2}-1$ (2)

• At $x^*_0=0$: $f'(0)=0-1=-1<0$ → stable (low "OFF" state) (1).
• At $x^*_-=2-\sqrt3\approx0.268$: $1+x^2\approx1.0718$, $(1+x^2)^2\approx1.149$; $8x\approx2.144$; $f'\approx2.144/1.149-1\approx0.866>0$ → unstable (threshold/separatrix) (1.5).
• At $x^*_+=2+\sqrt3\approx3.732$: $1+x^2\approx14.93$, squared $\approx222.9$; $8x\approx29.86$; $f'\approx29.86/222.9-1\approx-0.866<0$ → stable (high "ON" state) (1.5).

(d) (4 marks)

• The two stable states (ON/OFF) let a cell "remember" a transient stimulus after it's removed — hysteresis/memory (2).
• Bulk RNA-seq averages populations, reporting an intermediate mean that masks bimodality. Single-cell sequencing measures per-cell expression, revealing a bimodal distribution (two peaks) — the physical signature of coexisting states (2).

(e) (4 marks)

• Requires $n>1$ (cooperativity / ultrasensitivity) (2).
• Justification: with $n=1$ the production curve is hyperbolic and can intersect the degradation line $\gamma x$ at most once (besides origin), giving a single state. $n>1$ makes production sigmoidal so it can cross the straight degradation line up to three times → bistability geometrically requires the S-shape from cooperative binding (2).

Question 2

(a) (6 marks) Rows = metabolites $A,B$; columns = $v_1,v_2,v_3$: $S=\begin{pmatrix} +1 & -1 & 0\\ 0 & +2 & -1\end{pmatrix}$ (3) Steady state $S\mathbf v=0$:

• A: $v_1 - v_2 = 0 \Rightarrow v_1=v_2$ (1.5)
• B: $2v_2 - v_3 = 0 \Rightarrow v_3 = 2v_2 = 2v_1$ (1.5)

(b) (6 marks) Objective $Z=v_3=2v_1$, increasing in $v_1$ (2). Constraint $v_1\le10$ ⇒ optimum at $v_1=10$ (2). Optimal flux vector: $v_1=10,\ v_2=10,\ v_3=20$, giving $Z_{\max}=20$ (2).

(c) (4 marks)

• FBA imposes only mass-balance (stoichiometry) + flux bounds and optimizes an objective; concentrations' time-derivatives set to zero eliminate the need for rate constants/$K_m$ (2).
• Valid assumptions: (i) internal metabolite pools are at quasi-steady state (fast relative to growth/environmental change); (ii) cell is under an evolutionary optimality objective (e.g., maximal growth) — any two acceptable (2).

(d) (4 marks)

• Transcriptomics/proteomics indicate which enzymes are expressed → allow switching reactions off or bounding $v_i$ (integrating expression as flux constraints) (2).
• Metabolomics provides measured concentrations/exchange fluxes to further constrain uptake/secretion, resolving the many alternate-optima degeneracy of pure stoichiometric FBA (2).

Question 3

(a) (6 marks) Each stage: $g(u)=\frac{u^h}{k^h+u^h}$, a sigmoid of local steepness ~$h$. Composing $m$ stages, $R=g\circ g\circ\cdots\circ g(S)$ (2). Near the switch point the local logarithmic sensitivity multiplies: the effective Hill coefficient of a composition is approximately the product of stage sensitivities, so $h_{\text{eff}} \approx h^m$ (2). Thus a cascade of moderately cooperative steps produces a very steep (near-digital) response — cascades amplify ultrasensitivity, sharpening decision-making (2). (Accept $h_{\text{eff}}\sim m\,h$ style arguments if reasoned; full marks for multiplicative reasoning.)

(b) (6 marks)

• Spatial-only: e.g., mapping the boundary/gradient of gene expression across a tumor margin, or ligand–receptor signaling between physically adjacent cell types (niche interactions) — requires preserved coordinates (3).
• Dissociated-better: e.g., discovering rare cell types or full transcriptome-depth cell-type classification, where higher gene capture/sensitivity per cell (and unbiased whole-tissue sampling) outperforms spatial platforms (3).

(c) (4 marks)

• Scientific: unknown-function genes reveal gaps in our understanding of the minimal requirements for life; a chassis for engineering/defining core biology (2).
• Ethical/societal: biosafety/biosecurity concerns (engineered/synthetic organisms, dual-use), plus questions on ownership, containment, and "creating life" — need governance (2).

[
  {"claim":"Nonzero steady states of Q1(b) are 2±√3","code":"x=symbols('x'); sols=solve(Eq(4*x**2/(1+x**2)-x,0),x); vals=set(sols); result = ({0,2-sqrt(3),2+sqrt(3)}==vals)"},
  {"claim":"f'(x)=8x/(1+x^2)^2 -1 and f'(0)=-1 (stable), f'(2+√3)<0 (stable), f'(2-√3)>0 (unstable)","code":"x=symbols('x'); f=4*x**2/(1+x**2)-x; fp=diff(f,x); a=fp.subs(x,0); b=simplify(fp.subs(x,2+sqrt(3))); c=simplify(fp.subs(x,2-sqrt(3))); result = (a==-1) and (b<0) and (c>0)"},
  {"claim":"FBA optimum: v1=v2=10, v3=20, Zmax=20","code":"v1=10; v2=v1; v3=2*v2; result = (v2==10 and v3==20)"}
]