GATE Biotechnology Fundamentals of Biological Engineering Syllabus

What is the fundamental relation between standard Gibbs free energy of reaction and the equilibrium constant?

$$\Delta G^{\circ} = -RT \ln K$$ where $K$ is the thermodynamic equilibrium constant. A large negative $\Delta G^{\circ}$ corresponds to $K \gg 1$ (products favored).

Define chemical potential $\mu_i$ of a species in a mixture.

$$\mu_i = \left(\frac{\partial G}{\partial n_i}\right)_{T,P,n_{j\neq i}}$$ It is the partial molar Gibbs energy; equality of $\mu_i$ across phases defines phase equilibrium.

Define fugacity and how it relates to chemical potential for a real system.

Fugacity $f_i$ is an 'effective pressure' such that $$\mu_i = \mu_i^{\circ} + RT \ln\frac{f_i}{f_i^{\circ}}$$ For an ideal gas $f_i = P_i$. The fugacity coefficient $\phi_i = f_i / (y_i P)$ measures deviation from ideality.

Define activity and activity coefficient in solution thermodynamics.

Activity $a_i = \gamma_i x_i$, where $\gamma_i$ is the activity coefficient and $x_i$ the mole fraction. Chemical potential: $\mu_i = \mu_i^{\circ} + RT\ln a_i$. For an ideal solution $\gamma_i = 1$.

State the condition for phase equilibrium between two phases $\alpha$ and $\beta$.

For every species $i$: equal temperature, equal pressure, and equal chemical potential (or fugacity): $$\mu_i^{\alpha} = \mu_i^{\beta} \quad\Leftrightarrow\quad f_i^{\alpha} = f_i^{\beta}$$

State the Gibbs phase rule.

$$F = C - P + 2$$ where $F$ = degrees of freedom, $C$ = number of components, $P$ = number of phases. The 2 accounts for temperature and pressure.

State Raoult's law for an ideal liquid solution in vapor-liquid equilibrium.

$$p_i = x_i\, P_i^{sat}$$ The partial pressure of component $i$ in the vapor equals its liquid mole fraction times its pure-component saturation (vapor) pressure.

State Henry's law and where it applies.

$$p_i = H_i\, x_i$$ The partial pressure of a dilute solute is proportional to its liquid mole fraction; $H_i$ is Henry's constant. It applies to dilute (low concentration) solutes, e.g., gas solubility such as $\ce{O2}$ in water.

Write the equilibrium dissociation of a ligand-receptor complex and define $K_d$.

$$\ce{RL <=> R + L}, \qquad K_d = \frac{[R][L]}{[RL]}$$ $K_d$ (dissociation constant) is the free ligand concentration at which half the receptor sites are occupied; lower $K_d$ means higher affinity.

Write the fractional saturation (binding isotherm) for simple single-site ligand binding.

$$\theta = \frac{[L]}{K_d + [L]} = \frac{[L]}{[L] + K_d}$$ where $\theta$ is the fraction of occupied sites. At $[L]=K_d$, $\theta = 0.5$.

State the Hill equation for cooperative ligand binding and the meaning of the Hill coefficient.

$$\theta = \frac{[L]^{n}}{K_d + [L]^{n}}$$ The Hill coefficient $n$ indicates cooperativity: $n>1$ positive cooperativity, $n=1$ non-cooperative, $n<1$ negative cooperativity.

Write the Nernst equation for the equilibrium membrane potential of a single ion.

$$E_{ion} = \frac{RT}{zF}\ln\frac{[C]_{out}}{[C]_{in}}$$ where $z$ is ionic charge and $F$ is the Faraday constant. At $37^{\circ}\text{C}$ for a monovalent cation: $E \approx 61.5\,\log_{10}\frac{[C]_{out}}{[C]_{in}}$ mV.

Write the Goldman–Hodgkin–Katz (GHK) voltage equation for resting membrane potential.

$$V_m = \frac{RT}{F}\ln\frac{P_K[K^+]_o + P_{Na}[Na^+]_o + P_{Cl}[Cl^-]_i}{P_K[K^+]_i + P_{Na}[Na^+]_i + P_{Cl}[Cl^-]_o}$$ It accounts for the relative permeabilities $P$ of multiple ions.

Relate the standard free energy change to the standard reduction potentials in a redox reaction.

$$\Delta G^{\circ} = -nF\,\Delta E^{\circ}, \qquad \Delta E^{\circ} = E^{\circ}_{acceptor} - E^{\circ}_{donor}$$ A positive $\Delta E^{\circ}$ (electrons flow to higher reduction potential) gives negative $\Delta G^{\circ}$ (spontaneous).

How many ATP are generated per NADH and per FADH$_2$ via oxidative phosphorylation (standard P/O values)?

Approximately $2.5$ ATP per $\ce{NADH}$ (P/O $\approx 2.5$) and $1.5$ ATP per $\ce{FADH2}$ (P/O $\approx 1.5$). Older textbook values are 3 and 2 respectively.

State the net ATP and reducing-equivalent yield of glycolysis (glucose to 2 pyruvate).

Net per glucose: $+2$ ATP (4 produced $-$ 2 invested), $+2\ \ce{NADH}$, and 2 pyruvate. Overall: $\ce{glucose + 2 NAD+ + 2 ADP + 2 Pi -> 2 pyruvate + 2 NADH + 2 ATP + 2 H2O}$.

What is the total ATP yield from complete aerobic oxidation of one glucose molecule (textbook value)?

About $30$–$32$ ATP per glucose (glycolysis + pyruvate oxidation + TCA cycle + oxidative phosphorylation), depending on the shuttle used for cytosolic NADH and the assumed P/O ratios.

Distinguish a Newtonian from a non-Newtonian fluid.

A Newtonian fluid has shear stress directly proportional to shear rate with constant viscosity: $\tau = \mu\dfrac{du}{dy}$. A non-Newtonian fluid has a viscosity that varies with shear rate (apparent viscosity is not constant).

Write the power-law (Ostwald–de Waele) model and classify fluids by the flow behavior index $n$.

$$\tau = K\left(\frac{du}{dy}\right)^{n}$$ $n<1$: pseudoplastic (shear-thinning); $n=1$: Newtonian (then $K=\mu$); $n>1$: dilatant (shear-thickening). $K$ is the consistency index.

Define a Bingham plastic fluid.

A Bingham plastic requires a finite yield stress $\tau_0$ before it flows: $$\tau = \tau_0 + \mu_p\frac{du}{dy} \quad (\tau > \tau_0)$$ Below $\tau_0$ it behaves as a solid. Many mycelial broths and toothpaste-like fluids are Bingham.

Define the Reynolds number for pipe flow and state laminar/turbulent transition values.

$$Re = \frac{\rho u D}{\mu}$$ For pipe flow: laminar for $Re < 2100$, transitional $2100 < Re < 4000$, turbulent for $Re > 4000$.

Define the impeller Reynolds number for a stirred tank.

$$Re_i = \frac{\rho N D^{2}}{\mu}$$ where $N$ is impeller rotational speed (rev/s) and $D$ is impeller diameter. Laminar regime $Re_i < 10$; turbulent $Re_i > 10^{4}$.

What is the Hagen–Poiseuille equation for laminar flow in a pipe?

$$Q = \frac{\pi \Delta P\, R^{4}}{8\mu L}$$ The volumetric flow rate of laminar flow varies with the fourth power of the radius. Velocity profile is parabolic with $u_{max} = 2\bar{u}$.

Define the power number $N_p$ for an agitated vessel.

$$N_p = \frac{P}{\rho N^{3} D^{5}}$$ where $P$ is power input, $N$ impeller speed, $D$ impeller diameter. In the turbulent regime $N_p$ is approximately constant for a given impeller geometry.

How does ungassed power input depend on impeller speed and diameter in the turbulent regime?

With $N_p$ constant in turbulent flow: $$P = N_p\,\rho N^{3} D^{5}$$ Power scales with the cube of speed and the fifth power of impeller diameter.

Define mixing time in a bioreactor.

Mixing time $t_m$ is the time required to achieve a specified degree of homogeneity (typically 95%) after a tracer pulse is added. Shorter $t_m$ means better, faster blending. It is often correlated as $N\,t_m \approx$ constant for a given geometry in turbulent flow.

State Fick's first law of molecular diffusion.

$$J = -D\,\frac{dC}{dx}$$ The diffusive molar flux $J$ is proportional to the concentration gradient; $D$ is the diffusion coefficient (diffusivity). The minus sign indicates flux from high to low concentration.

State the key assumption of the film (two-film) theory of mass transfer.

All resistance to mass transfer resides in a thin stagnant fluid film at the interface, across which transport is by steady molecular diffusion; the bulk fluid is well mixed. The mass transfer coefficient $k = D/\delta$, where $\delta$ is the film thickness.

Write the equation for the rate of oxygen transfer (OTR) from gas to liquid in a bioreactor.

$$OTR = k_L a\,(C^{*} - C_L)$$ where $k_L a$ is the volumetric mass transfer coefficient, $C^{*}$ the saturation (equilibrium) dissolved $\ce{O2}$ concentration, and $C_L$ the bulk dissolved $\ce{O2}$ concentration.

At steady state in a fermenter, how do oxygen transfer rate (OTR) and oxygen uptake rate (OUR) relate?

$$OTR = OUR \implies k_L a\,(C^{*} - C_L) = q_{O_2}\,X$$ where $q_{O_2}$ is the specific oxygen uptake rate and $X$ the biomass concentration. Oxygen supply must match consumption.

Describe the static gassing-out (dynamic) method for measuring $k_L a$.

Stop aeration (or sparge $\ce{N2}$) to lower $C_L$, then resume aeration and record the dissolved $\ce{O2}$ rise. With no respiration, $\dfrac{dC_L}{dt} = k_L a(C^{*}-C_L)$; a plot of $\ln(C^{*}-C_L)$ vs $t$ gives slope $-k_L a$.

State Fourier's law of conduction and Newton's law of cooling (convection).

Conduction: $$q = -kA\frac{dT}{dx}$$ Convection: $$q = hA(T_s - T_\infty)$$ where $k$ is thermal conductivity, $h$ the convective heat transfer coefficient, $A$ the area.

Define the overall heat transfer coefficient $U$ for a wall with two convective films (neglecting wall curvature).

$$\frac{1}{U} = \frac{1}{h_i} + \frac{x_w}{k_w} + \frac{1}{h_o}$$ The total resistance is the sum of inside film, wall conduction, and outside film resistances. Then $q = U A\,\Delta T$.

Define the Log Mean Temperature Difference (LMTD) for a heat exchanger.

$$\Delta T_{lm} = \frac{\Delta T_1 - \Delta T_2}{\ln\!\left(\dfrac{\Delta T_1}{\Delta T_2}\right)}$$ where $\Delta T_1$ and $\Delta T_2$ are the temperature differences at the two ends. Heat duty: $q = U A\,\Delta T_{lm}$.

For the same inlet and outlet temperatures, which gives a larger LMTD: counter-current or co-current flow, and why?

Counter-current flow gives a larger LMTD and more uniform driving force, so it requires less area for the same duty. It can also cool/heat one stream beyond the other's outlet temperature, which co-current cannot.

Define heat exchanger effectiveness $\varepsilon$ in the $\varepsilon$-NTU method.

$$\varepsilon = \frac{q_{actual}}{q_{max}} = \frac{q}{C_{min}(T_{h,in} - T_{c,in})}$$ where $C_{min}$ is the smaller of the two fluid heat capacity rates $\dot{m}c_p$. It is the ratio of actual to maximum possible heat transfer.