Question

DNA microarrays or "chips" first appeared on the market in \(1996 .\) These chips are divided into square patches, with each patch having strands of DNA of the same sequence attached to a substrate. The patches are differentiated by differences in the DNA sequence. One can introduce DNA or mRNA of unknown sequence to the chip and monitor to which patches the introduced strands bind. This technique has a wide variety of applications in genome mapping and other areas. Modeling the chip as a surface with binding sites, and modeling the attachment of DNA to a patch using the Langmuir model, what is the required difference in the Gibbs energy of binding needed to modify the fractional coverage on a given patch from 0.90 to 0.10 for two different DNA strands at the same concentration at \(298 \mathrm{K} ?\) In performing this calculation replace pressure (P) with concentration (c) in the fractionalcoverage expression. Also, recall that \(K=\exp (-\Delta G / R T)\)

Short answer

The required difference in the Gibbs energy of binding to modify the fractional coverage on a given patch from 0.90 to 0.10 for two different DNA strands at the same concentration at 298K is approximately \(4305.78 \hspace{2mm}\)J/mol.

Step-by-step solution

Write down the given information

We are given: - Fractional coverage for strand 1 = 0.90 - Fractional coverage for strand 2 = 0.10 - Temperature, T = 298 K - Langmuir relationship: \(K = \exp (-\Delta G / RT)\) - Replace pressure (P) with concentration (c)

Express fractional coverage in terms of Gibbs energy

Using the Langmuir relationship, we can express the fractional coverage (θ) as: θ = \( \frac{cK}{1+cK}\) where K is the equilibrium constant related to the Gibbs energy: \( K = \exp (-\Delta G/RT) \)

Find expressions for Gibbs energy for each strand

For strand 1: θ1 = 0.90 θ1 = \( \frac{cK_1}{1+cK_1}\) For strand 2: θ2 = 0.10 θ2 = \( \frac{cK_2}{1+cK_2}\) Now we have two expressions for Gibbs energy related to each strand: \( K_1 = \exp (-\Delta G_1 / RT) \) \( K_2 = \exp (-\Delta G_2 / RT) \)

Solve for K1 and K2 using fractional coverage expressions

Solve the expressions for fractional coverage for K1 and K2: K1 = \( \frac{9c}{1 - 9c} \) K2 = \( \frac{c}{9 - c} \)

Determine the difference in Gibbs energy

The difference in the Gibbs energy of binding is: \(\Delta \Delta G = \Delta G_2 - \Delta G_1 = RT \ln(\frac{K_2}{K_1})\) Plug in values for K1 and K2 from Step 4: \(\Delta \Delta G = RT \ln(\frac{(c / (9 - c))}{(9c / (1 - 9c))})\) Notice that the 'c' terms cancel each other: \(\Delta \Delta G = RT \ln(\frac{1 - 9c}{9 - c} * 9)\)

Calculate the difference in Gibbs energy

Given the temperature T = 298K, and using the gas constant R = 8.314 J/(mol K), we can find the difference in Gibbs energy: \(\Delta \Delta G = (8.314)(298) \ln(\frac{1 - 9c}{9 - c} * 9)\) We now need to find the value of 'c' that satisfies the equation. Since the concentration will not affect the ratio between the fractional coverages of the two strands, we can set c=1/9: \(\Delta \Delta G = (8.314)(298) \ln(\frac{1 - 9(1/9)}{(9 - 1)/9} * 9)\) Calculate the ΔΔG: \(\Delta \Delta G = (8.314)(298) \ln(\frac{1}{8} * 9) \approx 4305.78 \hspace{2mm}\)J/mol So, the required difference in the Gibbs energy of binding to modify the fractional coverage on a given patch from 0.90 to 0.10 for two different DNA strands at the same concentration at 298K is approximately 4305.78 J/mol.

Key concepts

Langmuir model

The Langmuir model is a fundamental concept in surface chemistry that describes the adsorption of molecules on a solid surface. It assumes that the surface contains a finite number of binding sites that can be occupied by adsorbate molecules, and each site can hold only one molecule. The model considers that the adsorption and desorption of molecules are dynamic processes in equilibrium.

In the context of DNA microarrays, the Langmuir model is applied to understand how single strands of DNA or RNA bind to complementary DNA sequences affixed on the microarray surface. The equilibrium constant, K, in the Langmuir formula represents the binding affinity between the strands and is defined as the ratio of the rate constants for adsorption and desorption processes. Mathematically, it can be expressed as:\[\begin{equation}K = \exp(-\Delta G / RT)\end{equation}\]where \( \Delta G \) is the Gibbs energy of binding, R is the universal gas constant, and T is the absolute temperature. Understanding the Langmuir model is crucial for analyzing the binding interactions during DNA microarray experiments and interpreting the data accordingly.

Fractional coverage

Fractional coverage, often denoted by \( \theta \), is a term that signifies the proportion of the binding sites occupied by adsorbate on a surface at a given time. It's an essential measure in the Langmuir adsorption model and helps in understanding the extent to which DNA strands bind to microarray patches.

As it pertains to DNA microarrays, high fractional coverage indicates a strong hybridization between the test sample DNA and the DNA on the microarray. Conversely, low fractional coverage suggests weaker interaction. For instance, changing the fractional coverage from 0.90 to 0.10 signifies a substantial decrease in the binding affinity, which directly correlates to an increased difference in the Gibbs free energy (\( \Delta G \)) of binding. This relationship is pivotal when analyzing DNA microarray data, especially in measuring gene expression levels or the presence of mutations.

DNA microarray applications

DNA microarrays have revolutionized biology by enabling the simultaneous analysis of the expression levels of thousands of genes. This innovative technology has numerous applications across various fields of study.

For example, in biomedical research, they are used for gene expression profiling, which helps to understand diseases such as cancer and to discover new drug targets. Meanwhile, in clinical diagnostics, microarrays are instrumental for detecting genetic disorders through comparative genomic hybridization. In agriculture, they assist in crop improvement by analyzing the genetic variations of plants. Lastly, in environmental sciences, DNA microarrays are used to detect and monitor microorganisms in different ecosystems. The versatility of DNA microarray technology underpins its importance in advancing our comprehension of complex biological systems and improving health and environmental outcomes.

Genome mapping

Genome mapping refers to the process of locating and identifying the positions of genes or other genetic markers within the genome. This is a critical step in understanding the genetic basis of traits and diseases, and DNA microarrays serve as a powerful tool in this domain.

Using microarrays, scientists can perform large-scale studies that map the entire genome. This includes single nucleotide polymorphism (SNP) mapping, which helps identify genetic variations among individuals that may contribute to different traits or susceptibility to diseases. Microarrays are particularly useful in comparative genome studies, where one can look at genetic differences between species or strains. As a result, genome mapping using DNA microarrays accelerates our ability to decipher genetic codes and fosters the development of individualized medicine, tailored crop breeds, and other advances tailored by genetic research.