A student proposed the following hypothesis regarding the growth of a bacterial biofilm in… — ACT

A student proposed the following hypothesis regarding the growth of a bacterial biofilm in a flow chamber:

*Hypothesis*: The biofilm growth rate ( $G$ , in $\mu\text{m/day}$ ) is directly proportional to the bulk nutrient concentration ( $C$ , in $\text{mg/L}$ ) across all concentrations, and the rate of decrease in $G$ per unit increase in fluid shear stress ( $\tau$ , in $\text{Pa}$ ) is constant.

To test this hypothesis, the student conducted two experiments. In Experiment 1, the student varied $C$ while maintaining a constant $\tau$ of $0.10\text{ Pa}$ . In Experiment 2, the student varied $\tau$ while maintaining a constant $C$ of $10.0\text{ mg/L}$ . The results are shown in the tables below.

**Experiment 1 ( $\tau = 0.10\text{ Pa}$ )**

Bulk Nutrient Concentration ( $C$ , mg/L)	Biofilm Growth Rate ( $G$ , $\mu\text{m/day}$ )
2.0	0.5
4.0	1.0
8.0	2.0
16.0	2.0

**Experiment 2 ( $C = 10.0\text{ mg/L}$ )**

Fluid Shear Stress ( $\tau$ , Pa)	Biofilm Growth Rate ( $G$ , $\mu\text{m/day}$ )
0.05	2.4
0.10	2.0
0.20	1.5
0.40	1.0

Based on the results of Experiments 1 and 2, which of the following statements describes how the student should modify the hypothesis?

A
Biofilm growth rate ( $G$ ) is directly proportional to bulk nutrient concentration ( $C$ ) across all concentrations; and the rate of decrease in $G$ per unit increase in shear stress ( $\tau$ ) increases as $\tau$ increases.
Biofilm growth rate ( $G$ ) is directly proportional to bulk nutrient concentration ( $C$ ) only up to a threshold concentration, after which $G$ becomes independent of $C$ ; and the rate of decrease in $G$ per unit increase in shear stress ( $\tau$ ) decreases as $\tau$ increases.Cevap
C
Biofilm growth rate ( $G$ ) is independent of bulk nutrient concentration ( $C$ ) at low concentrations but becomes directly proportional at high concentrations; and the rate of decrease in $G$ per unit increase in shear stress ( $\tau$ ) remains constant.
D
Biofilm growth rate ( $G$ ) is directly proportional to bulk nutrient concentration ( $C$ ) only up to a threshold concentration, after which $G$ becomes independent of $C$ ; and biofilm growth rate ( $G$ ) increases as shear stress ( $\tau$ ) increases.

Cevap

Biofilm growth rate (

G

) is directly proportional to bulk nutrient concentration (

C

) only up to a threshold concentration, after which

G

becomes independent of

C

; and the rate of decrease in

G

per unit increase in shear stress (

\tau

) decreases as

\tau

increases.

The correct option states that the growth rate is directly proportional to the nutrient concentration up to a threshold, and that the rate of decrease per unit increase in shear stress decreases as shear stress increases. This is supported by Experiment 1, which shows a linear increase in growth rate from

0.5

2.0\text{ }\mu\text{m/day}

as nutrient concentration increases from

2.0

8.0\text{ mg/L}

, followed by a constant growth rate of

2.0\text{ }\mu\text{m/day}

at higher concentrations. Furthermore, Experiment 2 demonstrates that as shear stress increases, the growth rate decreases at a declining rate: the average rate of change decreases in magnitude from

-8.0

-5.0

, and then to

-2.5\text{ }\mu\text{m}/(\text{day}\cdot\text{Pa})

across successive intervals.

Adım Adım Çözüm

Analyze Experiment 1 to determine the relationship between nutrient concentration (

C

) and growth rate (

G

G

increases linearly with

C

from

0.5

2.0\text{ }\mu\text{m/day}

C

goes from

2.0

8.0\text{ mg/L}

. However, at

16.0\text{ mg/L}

G

remains at

2.0\text{ }\mu\text{m/day}

, indicating that the relationship is directly proportional only up to a threshold concentration of

8.0\text{ mg/L}

This establishes how the first part of the hypothesis should be modified to account for nutrient saturation.

Analyze Experiment 2 to determine the general trend between shear stress (

\tau

) and growth rate (

G

\tau

increases from

0.05

0.40\text{ Pa}

G

decreases from

2.4

1.0\text{ }\mu\text{m/day}

, confirming that growth rate decreases as shear stress increases.

This rule-out step eliminates any modifications suggesting a positive relationship between growth rate and shear stress.

Calculate the rate of change (slope) of

G

with respect to

\tau

over successive intervals to test the linearity of the decrease.

From

0.05\text{ Pa}

0.10\text{ Pa}

, the rate of change is

\frac{2.0 - 2.4}{0.10 - 0.05} = -8.0\text{ }\mu\text{m}/(\text{day}\cdot\text{Pa})

. From

0.10\text{ Pa}

0.20\text{ Pa}

, it is

\frac{1.5 - 2.0}{0.20 - 0.10} = -5.0\text{ }\mu\text{m}/(\text{day}\cdot\text{Pa})

. From

0.20\text{ Pa}

0.40\text{ Pa}

, it is

\frac{1.0 - 1.5}{0.40 - 0.20} = -2.5\text{ }\mu\text{m}/(\text{day}\cdot\text{Pa})

Calculating the rates of change over different intervals evaluates whether the rate of decrease is constant, increasing, or decreasing.

Compare the calculated slopes to evaluate how the rate of decrease changes.

The magnitude of the rate of decrease (slopes of

-8.0

-5.0

, and

-2.5

) becomes smaller as shear stress increases. This means the rate of decrease in growth rate per unit increase in shear stress decreases.

This determines the correct modification for the second part of the hypothesis.

Anahtar Kavram

Evaluating and modifying a hypothesis based on experimental results showing non-linear relationships and saturation thresholds.

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