Consider a chemostat. You wish to know the number of cells in the reactor and the fraction of the cells that are viable (i.e., alive as determined by ability to divide). a. Write an equation for viable cell number \(\left(n_{v}\right)\). Assume that $$ \mu_{\text {met,rep. }}=\frac{\mu_{\mathrm{m}, \mathrm{rep}} S}{K_{\text {s,rep }}+S}-k_{d}^{\prime} $$ where \(\mu_{\text {net,rep }}=\) net specific replication rate, \(\mu_{m, r e p}=\) maximum specific replication rate, and \(K_{d}=\) death rate. \(K_{s, \text { rep }}\) is the saturation parameter. b. Derive an expression for the value of \(S\) at steady state. c. Write the number balance in the chemostat on dead cells \(\left(n_{d}\right)\). d. Derive an expression for the fraction of the total population which are dead cells.

Step by step solution

01

Equation for Viable Cell Number

The viable cell number, denoted as \(n_v\), can be determined using the replication rate and death rate. Given the equation for the net specific replication rate \(\mu_{\text{net, rep}}\):\[ \mu_{\text {net,rep }} = \frac{\mu_{\mathrm{m}, \,\text{rep}} S}{K_{\text{s,\text{rep}}} + S} - k_{d}' \] we can write the equation for the viable cell number as: \[ n_{v} = n_{v0} e^{\mu_{\text{net,rep}} t} = n_{v0} e^{(\frac{\mu_{\mathrm{m}, \,\text{rep}} S}{K_{\text{s,\text{rep}}} + S} - k_{d}') t} \] Where \(n_{v0}\) is the initial number of viable cells and \(t\) is time.

02

Expression for \(S\) at Steady State

At steady state, the growth rate equals the dilution rate, denoted as \(D\). Therefore: \[ D = \frac{\mu_{\mathrm{m}, \,\text{rep}} S}{K_{\text{s,\text{rep}}} + S} - k_{d}' \] Solving for \(S\): \[ D + k_{d}' = \frac{\mu_{\mathrm{m}, \,\text{rep}} S}{K_{\text{s,\text{rep}} } + S} \] Rearrange to find \(S\):\[ S = \frac{D + k_{d}'}{\mu_{\mathrm{m}, \,\text{rep}} - (D + k_{d}')} K_{s,\text{rep}} \]

03

Number Balance on Dead Cells

To write the balance for dead cells \(n_d\), consider the rate of death and the outflow of dead cells: \[ \frac{dn_d}{dt} = n_v k_{d}' - Dn_d \] At steady state, \(\frac{dn_d}{dt} = 0\), so: \[ 0 = n_v k_{d}' - Dn_d \] Therefore, the steady state balance for dead cells is: \[ n_d = \frac{n_v k_{d}'}{D} \]

04

Fraction of Total Population that are Dead Cells

The total population of cells \(n_{tot}\) is the sum of dead \(n_d\) and viable \(n_v\) cells: \[ n_{\text{tot}} = n_v + n_d \] The fraction of dead cells \(f_d\) is given by: \[ f_d = \frac{n_d}{n_{\text{tot}}} \] Substituting \(n_d = \frac{n_v k_{d}'}{D}\): \[ f_d = \frac{\frac{n_v k_{d}'}{D}}{n_v + \frac{n_v k_{d}'}{D}} = \frac{ k_{d}'}{D + k_{d}'} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Viable Cell Number

Understanding the viable cell number, denoted as \(n_v\), in a chemostat involves calculating the balance between cell replication and cell death. A viable cell is one that is capable of division and therefore contributes to population growth.
The net specific replication rate \(\mu_{\text {net, rep}}\) is crucial here and is defined by: \[ \mu_{\text {net,rep }} = \frac{\mu_{\mathrm{m}, \, \text{rep}} S}{K_{\text{s,\text{rep}}} + S} - k_d' \] Here:

• \( \mu_{m, \, \text{rep}} \) stands for the maximum specific replication rate.
• \( S \) is the substrate concentration.
• \( K_{s, \text { rep }} \) is the saturation parameter.
• \( k_d' \) is the death rate parameter.

The viable cell number follows an exponential pattern over time, which is given by: \[ n_v = n_{v0} e^{\mu_{\text{net,rep}} t} \] Therefore, the initial cell number \( n_{v0} \) grows according to the net replication rate adjusted by the death rate.

Net Specific Replication Rate

The net specific replication rate \(\mu_{\text{net,rep}}\) is a critical measure in assessing cell growth within the chemostat. It represents the effective rate of cell replication after accounting for cell death.
This rate is impacted by the substrate concentration \(S\):
\[ \mu_{\text {net,rep }} = \frac{\mu_{\mathrm{m}, \, \text{rep}} S}{K_{\text{s, \text{rep}} }+S} - k_d' \]
Understanding each component:

• \(\mu_{m, \, \text{rep}}\) is the potential maximum replication rate.
• \(S\) is the substrate or nutrient concentration that cells consume for replication.
• \(K_{s, \text { rep }}\) is the substrate concentration at which the replication rate is half its maximum.

Essentially, the net specific replication rate determines how effectively the cell population grows, considering both nutrient availability and cell death.

Steady State Substrate Concentration

At steady state, the growth rate balances the dilution rate \(D\). Simply put, the number of cells entering the reactor equals the number leaving, plus any cells dying.
The steady-state equation is given by: \[ D = \frac{\mu_{\mathrm{m}, \, \text{rep}} S}{K_{\text{s, \text { rep}} } + S} - k_d' \]
To find the steady-state substrate concentration \(S\), we rearrange the equation:
\[ S = \frac{D + k_d'}{\mu_{\text{m}, \, \text{rep}} - (D + k_d')} K_{s, \text{rep}} \]
This equation allows us to calculate the concentration of the substrate that will maintain a steady population in the chemostat, considering the dilution and death rates.

Number Balance on Dead Cells

In a chemostat, dead cells must also be accounted for to understand overall cell dynamics.
The number balance on dead cells \(n_d\) involves the rate of cell death and the outflow of these dead cells:
\[ \frac{dn_d}{dt} = n_v k_d' - Dn_d \] Here, \(dn_d/dt\) is the rate change of dead cells over time. At steady state, this rate is zero, leading to:
\[ n_d = \frac{n_v k_d'}{D} \]
This balance equation shows that the number of dead cells is proportional to the viable cell number, the death rate, and inversely proportional to the dilution rate.

Fraction of Dead Cells in Population

To understand the health of the cell population, it's critical to determine the fraction of dead cells.
The total population \(n_{tot}\) is the sum of viable \(n_v\) and dead cells \(n_d\):
\[ n_{\text{tot}} = n_v + n_d \] The fraction of dead cells \(f_d\) is then:
\[ f_d = \frac{n_d}{n_{\text{tot}}} \] Substituting the expression for \(n_d\):
\[ f_d = \frac{\frac{n_v k_d'}{D}}{n_v + \frac{n_v k_d'}{D}} = \frac{k_d'}{D + k_d'} \]
This result indicates that the fraction of dead cells is directly influenced by the death rate and the dilution rate. A lower dilution rate or higher death rate increases the fraction of dead cells in the population.