# Calculations for Molecular Biology and Biotechnology

**A Guide to Mathematics in the Laboratory 2e**

**By**

- Frank Stephenson, Applied Biosystems, Foster City, California
- Frank Stephenson, Applied Biosystems, Foster City, California

Calculations for Molecular Biology and Biotechnology: A Guide to Mathematics in the Laboratory, Second Edition, provides an introduction to the myriad of laboratory calculations used in molecular biology and biotechnology. The book begins by discussing the use of scientific notation and metric prefixes, which require the use of exponents and an understanding of significant digits. It explains the mathematics involved in making solutions; the characteristics of cell growth; the multiplicity of infection; and the quantification of nucleic acids. It includes chapters that deal with the mathematics involved in the use of radioisotopes in nucleic acid research; the synthesis of oligonucleotides; the polymerase chain reaction (PCR) method; and the development of recombinant DNA technology. Protein quantification and the assessment of protein activity are also discussed, along with the centrifugation method and applications of PCR in forensics and paternity testing.

View full description### Audience

Any Biology student with an interest in Biotechnology or Molecular Biology and new Biotechnology technicians.

### Book information

- Published: June 2010
- Imprint: ACADEMIC PRESS
- ISBN: 978-0-12-375690-9

### Table of Contents

Chapter 1 Scientific Notation and Metric Prefixes

Introduction

1.1 Significant Digits

1.1.1 Rounding Off Significant Digits in Calculations

1.2 Exponents and Scientific Notation

1.2.1 Expressing Numbers in Scientific Notation

1.2.2 Converting Numbers from Scientific Notation to Decimal Notation

1.2.3 Adding and Subtracting Numbers Written in Scientific Notation

1.2.4 Multiplying and Dividing Numbers Written in Scientific Notation

1.3 Metric Prefixes

1.3.1 Conversion Factors and Canceling Terms

Chapter Summary

Chapter 2 Solutions, Mixtures, and Media

Introduction

2.1 Calculating Dilutions - A General Approach

2.2 Concentrations by a Factor of X

2.3 Preparing Percent Solutions

2.4 Diluting Percent Solutions

2.5 Moles and Molecular Weight - Definitions

2.5.1 Molarity

2.5.2 Preparing Molar Solutions in Water with Hydrated Compounds

2.5.3 Diluting Molar Solutions

2.5.4 Converting Molarity to Percent

2.5.5 Converting Percent to Molarity

2.6 Normality

2.7 pH

2.8 pKa and the Henderson - Hasselbalch Equation

Chapter Summary

Chapter 3 Cell Growth

3.1 The Bacterial Growth Curve

3.1.1 Sample Data

3.2 Manipulating Cell Concentration

3.3 Plotting OD550 vs. Time on a Linear Graph

3.4 Plotting the Logarithm of OD550 vs. Time on a Linear Graph

3.4.1 Logarithms

3.4.2 Sample OD550 Data Converted to Logarithm Values

3.4.3 Plotting Logarithm OD550 vs. Time

3.5 Plotting the Logarithm of Cell Concentration vs. Time

3.5.1 Determining Logarithm Values

3.6 Calculating Generation Time

3.6.1 Slope and the Growth Constant

3.6.2 Generation Time

3.7 Plotting Cell Growth Data on a Semilog Graph

3.7.1 Plotting OD550 vs. Time on a Semilog Graph

3.7.2 Estimating Generation Time from a Semilog Plot of OD550 vs. Time

3.8 Plotting Cell Concentration vs. Time on a Semilog Graph

3.9 Determining Generation Time Directly from a Semilog Plot of Cell Concentration vs. Time

3.10 Plotting Cell Density vs. OD550 on a Semilog Graph

3.11 The Fluctuation Test

3.11.1 Fluctuation Test Example

3.11.2 Variance

3.12 Measuring Mutation Rate

3.12.1 The Poisson Distribution

3.12.2 Calculating Mutation Rate Using the Poisson Distribution

3.12.3 Using a Graphical Approach to Calculate Mutation Rate from Fluctuation Test Data

3.12.4 Mutation Rate Determined by Plate Spreading

3.13 Measuring Cell Concentration on a Hemocytometer

Chapter Summary

References

Chapter 4 Working with Bacteriophages

Introduction

4.1 Multiplicity of Infection (moi)

4.2 Probabilities and Multiplicity of Infection (moi)

4.3 Measuring Phage Titer

4.4 Diluting Bacteriophage

4.5 Measuring Burst Size

Chapter Summary

Chapter 5 Nucleic Acid Quantification

5.1 Quantification of Nucleic Acids by Ultraviolet (UV) Spectroscopy

5.2 Determining the Concentration of Double-Stranded DNA (dsDNA)

5.2.1 Using Absorbance and an Extinction Coefficient to Calculate Double-Stranded DNA (dsDNA) Concentration

5.2.2 Calculating DNA Concentration as a Millimolar (mM) Amount

5.2.3 Using PicoGreenÂ® to Determine DNA Concentration

5.3 Determining the Concentration of Single-Stranded DNA (ssDNA) Molecules

5.3.1 Single-Stranded DNA (ssDNA) Concentration Expressed in ÎĽg/mL

5.3.2 Determining the Concentration of High-Molecular-Weight Single-Stranded DNA (ssDNA) in pmol/ÎĽL

5.3.3 Expressing Single-Stranded DNA (ssDNA) Concentration as a Millimolar (mM) Amount

5.4 Oligonucleotide Quantification

5.4.1 Optical Density (OD) Unit

5.4.2 Expressing an Oligonucleotideâ€™s Concentration in ÎĽg/mL

5.4.3 Oligonucleotide Concentration Expressed in pmol/ÎĽL

5.5 Measuring RNA Concentration

5.6 Molecular Weight, Molarity, and Nucleic Acid Length

5.7 Estimating DNA Concentration on an Ethidium Bromide-Stained Gel

Chapter Summary

Chapter 6 Labeling Nucleic Acids with Radioisotopes

Introduction

6.1 Units of Radioactivity - The Curie (Ci)

6.2 Estimating Plasmid Copy Number

6.3 Labeling DNA by Nick Translation

6.3.1 Determining Percent Incorporation of Radioactive Label from Nick Translation

6.3.2 Calculating Specific Radioactivity of a Nick Translation Product

6.4 Random Primer Labeling of DNA

6.4.1 Random Primer Labeling - Percent Incorporation

6.4.2 Random Primer Labeling - Calculating Theoretical Yield

6.4.3 Random Primer Labeling - Calculating Actual Yield

6.4.4 Random Primer Labeling - Calculating Specific Activity of the Product

6.5 Labeling 3â€™ Termini with Terminal Transferase

6.5.1 3â€™-end Labeling with Terminal Transferase - Percent Incorporation

6.5.2 3â€™-end Labeling with Terminal Transferase - Specific Activity of the Product

6.6 Complementary DNA (cDNA) Synthesis

6.6.1 First Strand cDNA Synthesis

6.6.2 Second Strand cDNA Synthesis

6.7 Homopolymeric Tailing

6.8 In Vitro Transcription

Chapter Summary

Chapter 7 Oligonucleotide Synthesis

Introduction

7.1 Synthesis Yield

7.2 Measuring Stepwise and Overall Yield by the Dimethoxytrityl (DMT) Cation Assay

7.2.1 Overall Yield

7.2.2 Stepwise Yield

7.3 Calculating Micromoles of Nucleoside Added at Each Base Addition Step

Chapter Summary

Chapter 8 The Polymerase Chain Reaction (PCR)

Introduction

8.1 Template and Amplification

8.2 Exponential Amplification

8.3 Polymerase Chain Reaction (PCR) Efficiency

8.4 Calculating the Tm of the Target Sequence

8.5 Primers

8.6 Primer Tm

8.6.1 Calculating Tm Based on Salt Concentration, G/C Content, and DNA Length

8.6.2 Calculating Tm Based on Nearest-Neighbor Interactions

8.7 Deoxynucleoside Triphosphates (dNTPs)

8.8 DNA Polymerase

8.8.1 Calculating DNA Polymeraseâ€™s Error Rate

8.9 Quantitative Polymerase Chain Reaction (PCR)

Chapter Summary

References

Further Reading

Chapter 9 The Real-time Polymerase Chain Reaction (RT-PCR)

Introduction

9.1 The Phases of Real-time PCR

9.2 Controls

9.3 Absolute Quantification by the TaqMan Assay

9.3.1 Preparing the Standards

9.3.2 Preparing a Standard Curve for Quantitative Polymerase Chain Reaction (qPCR) Based on Gene Copy Number

9.3.3 The Standard Curve

9.3.4 Standard Deviation

9.3.5 Linear Regression and the Standard Curve

9.4 Amplification Efficiency

9.5 Measuring Gene Expression

9.6 Relative Quantification - The Î”Î”CT Method

9.6.1 The 2-Î”Î”CT Method - Deciding on an Endogenous Reference

9.6.2 The 2-Î”Î”CT Method - Amplification Efficiency

9.6.3 The 2-Î”Î”CT Method - is the Reference Gene Affected by the Experimental Treatment?

9.7 The Relative Standard Curve Method

9.7.1 Standard Curve Method for Relative Quantitation

9.8 Relative Quantification by Reaction Kinetics

9.9 The R0 Method of Relative Quantification

9.10 The Pfaffl Model

Chapter Summary

References

Further Reading

Chapter 10 Recombinant DNA

Introduction

10.1 Restriction Endonucleases

10.1.1 The Frequency of Restriction Endonuclease Cut Sites

10.2 Calculating the Amount of Fragment Ends

10.2.1 The Amount of Ends Generated by Multiple Cuts

10.3 Ligation

10.3.1 Ligation Using Î»-Derived Vectors

10.3.2 Packaging of Recombinant Î» Genomes

10.3.3 Ligation Using Plasmid Vectors

10.3.4 Transformation Efficiency

10.4 Genomic Libraries - How Many Clones Do You Need?

10.5 cDNA Libraries - How Many Clones are Enough?

10.6 Expression Libraries

10.7 Screening Recombinant Libraries by Hybridization to DNA Probes

10.7.1 Oligonucleotide Probes

10.7.2 Hybridization Conditions

10.7.3 Hybridization Using Double-Stranded DNA (dsDNA) Probes

10.8 Sizing DNA Fragments by Gel Electrophoresis

10.9 Generating Nested Deletions Using Nuclease BAL 31

Chapter Summary

References

Chapter 11 Protein

Introduction

11.1 Calculating a Proteinâ€™s Molecular Weight from Its Sequence

11.2 Protein Quantification by Measuring Absorbance at 280 nm

11.3 Using Absorbance Coefficients and Extinction Coefficients to Estimate Protein Concentration

11.3.1 Relating Absorbance Coefficient to Molar Extinction Coefficient

11.3.2 Determining a Proteinâ€™s Extinction Coefficient

11.4 Relating Concentration in Milligrams Per Milliliter to Molarity

11.5 Protein Quantitation Using A 280 When Contaminating Nucleic Acids are Present

11.6 Protein Quantification at 205 nm

11.7 Protein Quantitation at 205 nm When Contaminating Nucleic Acids are Present

11.8 Measuring Protein Concentration by Colorimetric Assay - The Bradford Assay

11.9 Using Î˛-Galactosidase to Monitor Promoter Activity and Gene Expression

11.9.1 Assaying Î˛-Galactosidase in Cell Culture

11.9.2 Specific Activity

11.9.3 Assaying Î˛-Galactosidase from Purified Cell Extracts

11.10 Thin Layer Chromatography (TLC) and the Retention Factor (Rf)

11.11 Estimating a Proteinâ€™s Molecular Weight by Gel Filtration

11.12 The Chloramphenicol Acetyltransferase (CAT) Assay

11.12.1 Calculating Molecules of Chloramphenicol Acetyltransferase (CAT)

11.13 Use of Luciferase in a Reporter Assay

11.14 In Vitro Translation - Determining Amino Acid Incorporation

11.15 The Isoelectric Point (pI) of a Protein

Chapter Summary

References

Further Reading

Chapter 12 Centrifugation

Introduction

12.1 Relative Centrifugal Force (RCF) (g Force)

12.1.1 Converting g Force to Revolutions Per Minute (rpm)

12.1.2 Determining g Force and Revolutions Per Minute (rpm) by Use of a Nomogram

12.2 Calculating Sedimentation Times

Chapter Summary

References

Further Reading

Chapter 13 Forensics and Paternity

Introduction

13.1 Alleles and Genotypes

13.1.1 Calculating Genotype Frequencies

13.1.2 Calculating Allele Frequencies

13.2 The Hardy - Weinberg Equation and Calculating Expected Genotype Frequencies

13.3 The Chi-Square Test - Comparing Observed to Expected Values

13.3.1 Sample Variance

13.3.2 Sample Standard Deviation

13.4 The Power of Inclusion (Pi)

13.5 The Power of Discrimination (Pd)

13.6 DNA Typing and Weighted Average

13.7 The Multiplication Rule

13.8 The Paternity Index (PI)

13.8.1 Calculating the Paternity Index (PI) When the Motherâ€™s Genotype is not Available

13.8.2 The Combined Paternity Index (CPI)

Chapter Summary

References

Further Reading

Appendix A

Index