Foundation mathematics for biosciences bryson

Copyright © 2017 Pearson Education All rights reserved May not be reproduced in any form without permission from the publisher, except fair uses permitted under U.S or applicable copyright law Foundation Mathematics for Biosciences At Pearson, we have a simple mission: to help people make more of their lives through learning We combine innovative learning technology with trusted content and educational expertise to provide engaging and effective learning experiences that serve people wherever and whenever they are learning From classroom to boardroom, our curriculum materials, digital learning tools and testing programmes help to educate millions of people worldwide – more than any other private enterprise Every day our work helps learning flourish, and wherever learning flourishes, so people To learn more, please visit us at www.pearson.com/uk Foundation Mathematics for Biosciences First Edition Ela Bryson & Jackie Willis Harlow, England • London • New York • Boston • San Francisco • Toronto • Sydney • Dubai • Singapore • Hong Kong Tokyo • Seoul • Taipei • New Delhi • Cape Town • São Paulo • Mexico City • Madrid • Amsterdam • Munich • Paris • Milan Pearson Education Limited Edinburgh Gate Harlow CM20 2JE United Kingdom Tel: +44 (0)1279 623623 Web: www.pearson.com/uk First edition published 2017 (print and electronic) © Pearson Education Limited 2017 (print and electronic) The rights of Elzbieta Bryson and Jacqueline Willis to be identified as authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988 The print publication is protected by copyright Prior to any prohibited reproduction, storage in a retrieval system, distribution or transmission in any form or by any means, electronic, mechanical, recording or otherwise, permission should be obtained from the publisher or, where applicable, a licence permitting restricted copying in the United Kingdom should be obtained from the Copyright Licensing Agency Ltd, Barnard's Inn, 86 Fetter Lane, London EC4A 1EN The ePublication is protected by copyright and must not be copied, reproduced, transferred, distributed, leased, licensed or publicly performed or used in any way except as specifically permitted in writing by the publishers, as allowed under the terms and conditions under which it was purchased, or as strictly permitted by applicable copyright law Any unauthorised distribution or use of this text may be a direct infringement of the authors’ and the publisher’s rights and those responsible may be liable in law accordingly All trademarks used herein are the property of their respective owners The use of any trademark in this text does not vest in the author or publisher any trademark ownership rights in such trademarks, nor does the use of such trademarks imply any affiliation with or endorsement of this book by such owners Pearson Education is not responsible for the content of third-party internet sites ISBN: 978-0-273-77458-7 (print) 978-0-273-77462-4 (PDF) 978-1-292-12559-6 (ePub) British Library Cataloguing-in-Publication Data A catalogue record for the print edition is available from the British Library Library of Congress Cataloging-in-Publication Data Names: Bryson, Elzbieta | Willis, Jackie Title: Foundation mathematics for biosciences / Elzbieta Bryson, Jacqueline  Willis Description: First edition | Harlow : Pearson, 2016 | Includes index Identifiers: LCCN 2016025414 | ISBN 9780273774587 | ISBN 9781292125596 (epub) Subjects: LCSH: Biomathematics Classification: LCC QH323.5 B79 2016 | DDC 570.1/51—dc23 LC record available at https://lccn.loc.gov/2016025414 10 21 20 19 18 17 Print edition typeset in Times NR MT Pro 10/12 by Lumina Datamatics, Inc Printed and Bound in Malaysia NOTE THAT ANY PAGE CROSS REFERENCES REFER TO THE PRINT EDITION Dedication To my wonderful Mum, Ela This page intentionally left blank Contents Preface Guided tour About the authors Acknowledgements viii x xii xiii 20 38 55 70 91 119 154 183 215 244 267 10 11 12 Basic arithmetic skills Fractions and decimals Units of measurement Ratios and percentages Logarithms Concentrations and dilutions Measurements in biology Analytical biology Molecular biology Enzyme kinetics Statistical calculations Graphs, trendlines and equations Answers 308 Appendix Arithmetic operations and commonly used Greek letters 351 Appendix Periodic table 353 Appendix Statistical tables 354 Appendix Correlation and linear regression analysis using Excel 356 Index 363   vii Preface Whether you have already purchased this book or are still contemplating buying it, we hope you will take some time reading this preface so that you can understand why this book was written and how to get the most out of it The Purpose of this Book The authors have spent many years supporting students with the mathematical demands of undergraduate and postgraduate courses in the biosciences We believe that you will benefit from our experience and the immense effort that we have poured into this book so that you become successful in both your degree course and future career Content This book consists of twelve chapters and each chapter is divided into two sections It is designed to allow you progress in a logical manner from sets of easier, fundamental problems to much more demanding and complex calculations aligned to various disciplines in biology In the first five chapters we cover the essential ground rules to enable a smooth transition into the later chapters We begin with the arithmetic operations in mathematics in Chapter 1, giving emphasis to the use of equations and indices In Chapter we move on to fractions and here you will also learn about the rounding of numbers and scientific notation Chapter introduces the SI units of measurement and rules for their use and conversions between viii  different units Ratios and percentages are discussed in Chapter 4, providing examples of calculations encountered when preparing mixtures and solutions with a given percentage concentration Chapter is dedicated to logarithms, giving clear explanations of the laws of logarithms and the application of logarithms in the biosciences In Chapter you will learn about preparing molar solutions and both standard and serial dilutions We know this is a problem area for many students, hence our decision to devote a whole chapter to these topics Chapters 7-10 present calculations relevant to the specialisms in biosciences Each chapter provides a brief overview of some of the theoretical concepts of each topic before working through typical calculations Chapter covers measurements made in microscopy, cell biology and microbiology as well as calculations of selected physiological and pharmacological parameters Chapter focuses on calculations relating to a range of techniques used in analytical biology and radiobiology Chapter contains examples of solutions to problems in DNA and protein analysis, whilst Chapter 10 is devoted to enzyme kinetics, including analysis of enzyme inhibition In Chapter 11 you are introduced to statistics and will conduct some statistical analysis Chapter 12 demonstrates how to present data correctly in graphs and charts as well as explore relationships between variables using correlation and regression analysis Preface Key Features • Learning Outcomes A summary is provided at the start of each chapter of the learning outcomes expected to be achieved once the chapter has been completed This will help you keep track of what you have learnt • Worked Examples Throughout the book there are numerous worked examples with detailed solutions and explanations, taking you step by step through each calculation • SELF-ASSESSMENT There are also calculations for you to attempt independently, then check against the answer key at the end of the book This will help you check your understanding and increase confidence as problems become progressively more difficult • MyMathLabGlobal This book is available with access to the online resource, MyMathLabGlobal, but requires that a course ID has been set up by your tutor for you to use it This e-resource provides an extensive bank of exercises developed by the authors to provide the opportunity for further self-assessment (examples of these questions are listed at the end of each half chapter of the book) MyMathLabGlobal will guide you through each step in solving a problem until the fully worked correct answer is displayed Your tutor has the option to set up homework, quizzes and tests • Key Terms Key terms are defined in each chapter and these are highlighted in coloured text where they are explained A list of key terms is also given at the end of the chapter, indicating those which may appear as a key term in other chapters of the book Reviewing the key terms once a chapter is completed will ensure you fully understand each concept and are ready to progress further In the event that Pearson invite us to produce a second edition, we would like to hear your suggestions on any improvements or additional material that could be included We can be contacted at: mathsforbiosciences@gmail.com Thank you for purchasing this book, we hope you will enjoy using it Ela Bryson Jackie Willis ix  www.downloadslide.com Appendix 3  Statistical tables Critical values of the t-distribution (two-tailed) df 354  Level of significance A 0.1 0.05 0.02 0.01 0.002 0.001 6.314 12.706 31.821 63.657 318.31 636.62 2.920 4.303 6.965 9.925 22.327 31.598 2.353 3.182 4.541 5.841 10.214 12.924 2.132 2.776 3.747 4.604 7.173 8.610 2.015 2.571 3.365 4.032 5.893 6.869 1.943 2.447 3.143 3.707 5.208 5.959 1.895 2.365 2.998 3.499 4.785 5.408 1.860 2.306 2.896 3.355 4.501 5.041 1.833 2.262 2.821 3.250 4.297 4.781 10 1.812 2.228 2.764 3.169 4.144 4.587 11 1.796 2.201 2.718 3.106 4.025 4.437 12 1.782 2.179 2.681 3.055 3.930 4.318 13 1.771 2.160 2.650 3.012 3.852 4.221 14 1.761 2.145 2.624 2.977 3.787 4.140 15 1.753 2.131 2.602 2.947 3.733 4.073 16 1.746 2.120 2.583 2.921 3.686 4.015 17 1.740 2.110 2.567 2.898 3.646 3.965 18 1.734 2.101 2.552 2.878 3.610 3.922 19 1.729 2.093 2.539 2.861 3.579 3.883 20 1.725 2.086 2.528 2.845 3.552 3.850 21 1.721 2.080 2.518 2.831 3.527 3.819 22 1.717 2.074 2.508 2.819 3.505 3.792 2.807 3.485 3.767 23 1.714 2.069 2.500 24 1.711 2.064 2.492 2.797 3.467 3.745 25 1.708 2.060 2.485 2.787 3.450 3.725 26 1.706 2.056 2.479 2.779 3.435 3.707 3.421 3.690 27 1.703 2.052 2.473 2.771 28 1.701 2.048 2.467 2.763 3.408 3.674 29 1.699 2.045 2.462 2.756 3.396 3.659 30 1.697 2.042 2.457 2.750 3.385 3.646 www.downloadslide.com Appendix • Statistical tables Critical values of the Chi-square distribution df Level of significance A 0.1 0.05 0.02 0.01 0.005 2.706 3.841 5.024 6.635 7.879 4.605 5.991 7.378 9.210 10.597 6.251 7.815 9.348 11.345 12.838 7.779 9.488 11.143 13.277 14.860 9.236 11.070 12.832 15.086 16.750 10.645 12.592 14.449 16.812 18.548 12.017 14.067 16.013 18.475 20.278 13.362 15.507 17.535 20.090 21.955 14.684 16.919 19.023 21.666 23.589 10 15.987 18.307 20.483 23.209 25.188 11 17.275 19.675 21.920 24.725 26.757 12 18.549 21.026 23.336 26.217 28.300 13 19.812 22.362 24.736 27.688 29.819 14 21.064 23.685 26.119 29.141 31.319 15 22.307 24.996 27.488 30.578 32.801   355 www.downloadslide.com Appendix 4 Correlation and linear regression analysis using Excel In order to use Excel for correlation and linear regression analysis, you need to install the Analysis ToolPak Installation of Analysis ToolPak From the File drop down menu, select Options and then click on Add-Ins This will open the following window: Select ‘Manage Excel Add-Ins’ and click on ‘Go…’ The following window will appear where you have to tick the box next to Analysis ToolPak and click ‘OK’ to confirm 356  www.downloadslide.com Appendix • Correlation and linear regression analysis using Excel The ToolPak should now be installed and, under the Data option, the Data Analysis tab will be visible on the right-hand side of the toolbar: You will now be able to perform correlation and linear regression analysis as shown below Correlation analysis We will use data from Worked example 12.2.1, in which correlation coefficient is calculated to find out if there is a linear relationship between the germination of seeds and the amount of monthly rainfall First, enter the data into two columns of an Excel spreadsheet: Now click on Data Analysis and select ‘Correlation’ from the drop down menu:   357 www.downloadslide.com Appendix • Correlation and linear regression analysis using Excel In the next step, specify the input range by selecting the data using a cursor If you include the cells with the variable names, tick the box ‘Labels in First Row’ After ‘OK’ is selected, the results will appear in a new sheet: We can see that there is a strong positive correlation between seed germination and rainfall as r = 0.973 (3 d.p.) We have obtained an identical value performing the calculations manually in Worked example 12.2.1 The above screenshot also shows that there is a perfect correlation between rainfall and rainfall and also between germination and germination, with values of r equal to 1, as one would expect 358  www.downloadslide.com Appendix • Correlation and linear regression analysis using Excel Linear regression analysis using Excel We will use data from Worked example 12.2.4, in which an equation of the calibration curve for protein concentration standards is obtained using linear regression First, enter the data into two columns of an Excel spreadsheet: Then, under ‘Data Analysis’, select ‘Regression’ from the list of Analysis Tools   359 www.downloadslide.com Appendix • Correlation and linear regression analysis using Excel In the Input Y range and Input X range, select the cells that contain the data for absorbance and protein concentration, respectively If you include the cells with the variable names, tick the box ‘Labels’ After ‘OK’ is selected, the results will appear in a new sheet They include the value of the gradient, m (labelled Concentration) and vertical intercept, c (labelled Intercept) If you did not include the cells with variable names in the data input range, then the gradient would be labelled as X-Variable Summary output Regression Statistics Multiple R 0.998496 R Square 0.996994 Adjusted R Square 0.996694 Standard Error 0.015267 Observations 12 Anova df Regression SS MS 0.773166 0.773166 Residual 10 0.002331 0.000233 Total 11 0.775497 360  F 3317.115 Significance F 6.0437E-14 www.downloadslide.com Appendix • Correlation and linear regression analysis using Excel Coefficients Intercept Concentration Standard Error t Stat P-value Lower 95 % Upper 95 % Lower 95.0 % Upper 95.0 % –0.018533 0.010050 –1.844108 0.094958 –0.040926 0.003860 –0.040926 0.003860 1.486286 0.025806 57.594402 0.000000 1.543785 1.543785 1.428786 1.428786   361 www.downloadslide.com This page intentionally left blank www.downloadslide.com Index A abscissa, 267 absolute zero, 51 absorbance, 155–161 acid dissociation constant (Ka), 82–83, 204–205, see also pKa adsorption chromatography, 169–171 affinity chromatography, 172–175 agarose gel electrophoresis, 195–198 migration distance, 195–198, 295–296 alveolar dead space, 140–141 alveolar ventilation, 140–141 amplicon, 185 amplification, magnitude of, 185–188 analyte, 154 anatomical dead space, 140–141 antilogarithm, 71 app apparent maximum velocity (Vmax ), 233–238 app apparent Michaelis constant (Kmax ), 229–237 approximation, 28–31 arithmetic mean, see mean arithmetic operations, 1, 351 atomic mass, 43, see also relative atomic mass atomic number, 43 average, see mean Avogadro constant, 91 Avogadro number, 39, 91 B bar chart, 276–278 base of indices, 5–7, 70 base of logarithms, 70–77 base units, 38–39 becquerel, 161–164 Beer–Lambert law, 156–159, 224–225 binding to enzyme, 238–240, see also Hill plot cooperative, 238–240 noncooperative, 238–239 BMI (body mass index), 39,143–144 BODMAS, 9–11 body mass index (BMI), 39, 143–144 body surface area, 145 buffer, 83–84 C calibration curve, 159–161, 175–177, 291–296, see also regression analysis Carbon and Clarke formula, 194–195 carbon dating, 87, see also radioisotope cardiac function, 141–143 cardiac output, 141–143 heart rate, 141–143 stroke volume, 141–143 Cartesian coordinate system, 267 abscissa, 267 ordinate, 267 origin, 267 catalytic constant (kcat), 225 cell counting, 129–134 haemocytometry, 129–133 pour plate technique, 133–134 viable cell count, 133–134 cell culture, 125–134, see also exponential growth confluency, 128–129 viability, 132 Celsius scale, 51–52 central tendency, see measures of central tendency charts, see also graphs bar chart, 276–278 histogram, 250–251, 278–279 pie chart, 57, 275–276 Chi-square test, 256–262 contingency table, 259–262 goodness of fit, 258–259 table of critical values, 357 Yates’ correction, 261–262 chromatography, 167–178 adsorption chromatography, 169–171 affinity, 172–175 high performance liquid chromatography (HPLC), 175–178, 293–294 ion exchange, 172, 174, 208 recovery, 170–171 resolution (Rs), 169–170 retention factor, 168–169 retention time, 169–170 size exclusion, 171–172 thin layer chromatography (TLC), 167–169 yield, 170, 172 class interval, 250–251 coefficient of variation, 248–249 common logarithms, 71–72 competitive inhibitors, 228–232 concentration mass, 97–99 molar, see molarity parts per billion (ppb), 97, 99–100 parts per million (ppm), 97, 99–100 percentage, 66–67, 91 confidence intervals, 250–253 confidence level, 251–253 363 www.downloadslide.com Index confidence limits, 251–253 confluency, 128–129 confounding factor, 286 conjugate base, 82 contingency table, 259–262, see also Chi-square test cooperative binding, 238–240 cooperativity, 238–240, see also Hill plot negative, 238–239 positive, 238–239 correlation, 284–290 negative, 285 positive, 285 correlation analysis, 284–290 using Excel, 356–358 correlation coefficient, 284–290 counting efficiency, 163–164 critical value, 252, 354–355 curie, 162 D dalton (Da), 43 data, 1, 244 categorical, 244, 256 continuous, 244 discrete, 1, 244, 246 qualitative, quantitative, transformation, 297–302 datum, see data decay constant, 86–87, 164–165 decimal number, 28 decimal places, 28–29 degree of inhibition, 228 degrees of freedom, number of, 252, 258–259, 261–263 denominator, 20 common, 22 derived units, 39–40 descriptive statistics, 244–253 measures of central tendency, 244–246 measures of dispersion, 246–253 diluent, 104 dilution factor, 103–106, 111–114 dilution fold, see dilution factor dilution formula, 106–110, 129 dilutions, 102–114, see also dilution factor; dilution formula serial, 111–114 standard, 103–110 dimensionless quantity, 40, 44, 45, 103, 121, 155 direct UV method, 156–159 dispersion, see measures of dispersion dissociation constant, 230, 232–240 DNA analysis, 183–198 agarose gel electrophoresis, 195–198 364 genomic libraries, 193–195 polymerase chain reaction (PCR), 184–189 quantification, 183–184 restriction endonuclease analysis, 189–193, 295–296 sequencing, 185–186, 188–189 DNA quantification, 183–184 DNA sequencing, 185–186, 188–189 melting temperature, 185–186, 188 primer, 185–186, 188 template DNA, 185–186 double reciprocal plot, see Lineweaver–Burk plot doubling time, see generation time drug dosing, 144–145 drug elimination, 86, 145–149, 298–302 elimination rate constant (kel), 86, 146–148, 298–302 first order kinetics, 146–147 half-life, 86, 147–148, 298–302 renal clearance, 148–149 zero order kinetics, 148 drug potency, 149–150 Du Bois equation, 145 E effective concentration at 50 % (EC50), 149–150 efficiency of PCR, 185–188 electric charge of proteins, 202–208, see also pI elimination rate constant (kel), 86, 146–148, 298–302 endonuclease, see restriction endonuclease enzyme cooperativity, see cooperativity enzyme inhibition, 228–238, see also inhibitors apparent maximum velocity (Vapp max), 233–238 apparent Michaelis constant (Kapp max), 229–237 degree of inhibition, 228 dissociation constant, 230, 232–238 enzyme inhibitors, see inhibitors enzyme kinetics, 215–240 catalytic constant (kcat), 225 enzyme cooperativity, 238–240 enzyme inhibition, 228–238 hyperbolic plot, 219–220 initial reaction rate (v0), 216–223 Lineweaver–Burk plot, 220–224, 296–297 maximum velocity (Vmax), 218–225 Michaelis constant (Km), 219–225 Michaelis–Menten equation, 219 Michaelis–Menten model, 215–225 equation definition of, 3, 11 Du Bois, 145 Henderson–Hasselbalch, 83–84, 205–206 Hill, 238–240 Michaelis–Menten, 219 www.downloadslide.com Index of a straight line, 14–16, 269–274, see also straight line of line of best fit, 160, 290–291, see also line of best fit rearrangement, 12–16 solving of, 11–16 substitution, 11 estimation, expected frequency, 256 exponent, see indices exponential decay, 86–87, 145–149, 298–302 decay constant, 86–87, 164–165 elimination rate constant (kel), 86, 146–148 half-life, 86–87, 147–148, 164–165, 298–302 exponential growth, 84–85, see also cell culture generation time, 85, 127 growth constant, 84–85 extrapolation, 292–293 F factor common, 20 highest common, 21 formula, see equation fractions, 20–26 arithmetic operations, 22–25 common, 20 decimal, 28 denominator, 20 equivalent, 20 improper, 20 numerator, 20 proper, 20 simple, 20 simplified, 20, 21 vulgar, 20 frequency, 250–251, 278–279 expected, 256 observed, 256 G generation time, 85, 127 genomic libraries, 193–195 Carbon and Clark formula, 194–195 goodness of fit test, 258–259 gradient, 14, 269–274 graphs, see also charts line graph, 280–281 scatterplot, 285–289 graticule, 121–124 calibration, 121–124 Greek letters commonly used, 352 growth constant, 84–85 H haemocytometer, 129–133 half-life of drug, 86–87, 147–148, 298–302 of radioisotope, 86–87, 164–165 heart rate, 141–143 Henderson–Hasselbalch equation, 83–84, 205–206 high performance liquid chromatography (HPLC), 175–178, 293–294 Hill coefficient, 238–240 Hill plot, 238–240 histogram, 250–251, 278–279 hyperbolic plot, 219–220 hypothesis alternative (H1), 255–256 null (H0), 255–256 I index, see indices indices, 5–9 laws of, 6–9 inhibitors, 228–238, see also enzyme inhibition competitive, 228–232 mixed, 235–238 noncompetitive, 236–238 uncompetitive, 232–235 initial reaction rate (v0), 216–223 integer, intercept horizontal, 268 vertical, 14, 268–274 interconversion of units, 47–53 International System of Units, see SI system interpolation, 292 ion exchange chromatography, 172, 174, 208 isoelectric point (pI), 208, see also electric charge of proteins isotope, 43–45, see also radioisotope K Ka (acid dissociation constant), 82–83, 204–205, see also pKa kcat (catalytic constant), 225 kel (elimination rate constant), 86, 146–148, 298–302 Km (Michaelis constant), 219–225 Kelvin scale, 51–52 L laws of indices, 6–9 logarithms, 73–76 365 www.downloadslide.com Index leading zeros, 29–30 least squares method, 290–291 line graph, 280–281 line of best fit, 160, 290–291 Lineweaver–Burk plot, 220–224, 296–297, see also reciprocal logarithms, 70–77 base of, 70 change of base, 76–77, 187–188 common, 71–72 definition of, 70 laws of, 73–76 natural, 75–77 lowest common multiple, 22–23 lung function, 138–141 alveolar dead space, 140–141 alveolar ventilation, 140–141 anatomical dead space, 140–141 physiological dead space, 140 respiratory cycle, 138 tidal volume, 138–141 ventilation rate, 138–141 M magnification, 120–125 mass concentration, 97–99 maximum velocity (Vmax), 218–225 mean, 2–3, 244–245 measures of central tendency, 244–246 mean, 2–3, 244–245 median, 245–246 mode, 246 measures of dispersion, 246–253 coefficient of variation, 248–249 confidence intervals, 250–253 range, 246–247 sample variance, see variance standard deviation, 247–248 standard error of the mean (SEM), 249–252 variance, 247 median, 245–246 melting temperature, 185–186, 188 Michaelis constant (Km), 219–225 Michaelis–Menten equation, 219 Michaelis–Menten model, 215–225, see also enzyme kinetics microscopy, 119–125 graticule, 121–124 magnification, 120–125 stage micrometer, 122–123 migration distance in DNA electrophoresis, 195–198, 295–296 in protein electrophoresis, 208–211, 294–295 366 mixed inhibitors, 235–238 mixed number, 20 mode, 246 molar concentration, see molarity molar extinction coefficient, 156–159 molar mass, 45, 92, see also relative molar mass molarity, 43, 91–100 and other types of concentrations, 97–100 mole, 39, 91 molecular mass, 44–45, see also relative molecular mass molecular weight, see relative molecular mass N natural logarithms, 75–77 noncompetitive inhibitors, 236–238 noncooperative binding, 238–239 normal distribution, 250–251 numerator, 20 O observed frequency, 256 One-sample Student t-test, 262–263 order, see indices ordinate, 267 origin, 267 P palindromic recognition sequence, 189–190, 192–193 parts per billion (ppb), 97, 99–100 parts per million (ppm), 97, 99–100 path length, 155–159 percentage change, 64–65 composition, 66–67 concentration, 66–67, 91 percentages, 62–67 periodic table, 353 pH, 79–82 definition of, 79 scale, 79–80 physiological dead space, 140 pI (isoelectric point), 208, see also electric charge of proteins pie chart, 57, 275–276 pKa, 82–84, 204–207, see also acid dissociation constant (Ka) pOH, 80–81 polyacrylamide gel electrophoresis, 208–211 migration distance, 208–211, 294–295 www.downloadslide.com Index polymerase chain reaction (PCR), 184–189 amplicon, 185 amplification, magnitude of, 185–188 efficiency, 185–188 melting temperature, 185–186, 188 primer, 184–186, 188–189 template DNA, 184–186 population, 244 population mean, 244–245, 262–263 pour plate technique, 133–134 power, see indices ppb (parts per billion), 97, 99–100 ppm (parts per million), 97, 99–100 prefixes, 40–41 change of, 47–50 prime factorisation, 21 prime number, 21 primer, 184–186, 188–189 probability, 256 product of multiplication, product of reaction, 215 proportions, 58–60 cross multiplication, 59–60 cross products, 59 protein analysis, 202–211 electric charge, 202–208 polyacrylamide gel electrophoresis, 208–211 Q quantification of DNA, 183–184 quotient, R radioactivity becquerel, 161–164 counting efficiency, 163–164 curie, 162 specific activity, 164 radioisotope, 86–87, 161–165 decay constant, 86–87, 164–165 half-life, 86–87, 164–165 range, 246–247 ratios, 55–58 equivalent, 55 rearrangement, 12–16 reciprocal, 25–26, 220 recognition sequence, 189–193 palindromic, 189–190, 192–193 recovery, 170–171 regression analysis, 290–302 using Excel, 356–357, 359–361 relationship causal, 286 non-causal, 286 relative atomic mass, 44, see also atomic mass relative molar mass, 45–46, see also molar mass relative molecular mass, 45, see also molecular mass; molecular weight reliability coefficient, 251–252 renal clearance, 148–149 resolution (Rs), 169–170 respiration, see lung function respiratory cycle, 138 restriction endonuclease, 189–193 analysis, 189–193, 295–296 recognition sequence, 189–193 retention factor, 168–169 time, 169–170 rounding decimal places, 28–29 of mathematical operations, 30–31 significant figures, 29–30 S sample, 244 sample mean, see mean sample variance, see variance scatterplot, 285–289 scientific calculator, 16–17 scientific notation, 32–35 use of scientific calculator, 35 SEM (standard error of the mean), 249–252 sequencing, see DNA sequencing SI system, 38–46 base quantities, 38–39 base units, 38–39 derived units, 39–40 prefixes, 40–41, 47–50 rules of use, 41–42 unit symbols, 39–40 units outside, 42–46 significance level, 252 significance testing, 256–263 Chi-square test, 256–262 goodness of fit test, 258–259 One-sample Student t-test, 262–263 probability, 256 significant figures, 29–30 size exclusion chromatography, 171–172 slope, see gradient solute, 66, 91 367 www.downloadslide.com Index solutions preparation of, 66–67, 92–100 stock, 102 specific activity, enzyme, 173–174 specific activity, radioisotope, 164 spectrophotometry, 154–161 absorbance, 155–161 Beer–Lambert law, 156–159, 224–225 direct UV method, 156–159 molar extinction coefficient, 156–159 path length, 155–159 transmittance, 40, 155, see also absorbance stage micrometer, 122–123 standard curve, see calibration curve standard deviation, 247–248 standard error, 249–252 standard error of the mean (SEM), 249–252 statistical hypotheses, 255–256 statistical tables, 354–355 statistical testing, 255–263 significance testing, 256–263 statistical hypotheses, 255–256 straight line equation of, 14–16, 269–274 gradient, 14, 269–274 horizontal intercept, 268 vertical intercept, 14, 268–274 stroke volume, 141–143 Student t-test, 262–263 substitution, 11 substrate, 215 sum, 2–3, 244, 351 T t-distribution, 251–252 table of critical values, 354 template DNA, 184–186 test, see also significance testing one-tailed, 256 two-tailed, 256 368 thin layer chromatography (TLC), 167–169 tidal volume, 138–141 trailing zeros, 29–30 transmittance, 40, 155, see also absorbance triple point of water, 51 t-test, 262–263 turnover number, see catalytic constant (kcat) U uncompetitive inhibitors, 232–235 unified atomic mass unit, see dalton (Da) unit prefixes, 40–41, 47–50 unit symbols, 39–40 V v0 (initial reaction rate), 216–223 Vmax (maximum velocity), 218–225 variance, 247 velocity app apparent maximum (Vmax ), 233–238 maximum (Vmax), 218–225 ventilation rate, 138–141 viability, 132 viable cell count, 133–134 Y Yates’ correction, 261–262 yield, 170, 172 Z zeros leading, 29–30 trailing, 29–30 ... record for the print edition is available from the British Library Library of Congress Cataloging-in-Publication Data Names: Bryson, Elzbieta | Willis, Jackie Title: Foundation mathematics for biosciences. .. flourishes, so people To learn more, please visit us at www.pearson.com/uk Foundation Mathematics for Biosciences First Edition Ela Bryson & Jackie Willis Harlow, England • London • New York • Boston... May not be reproduced in any form without permission from the publisher, except fair uses permitted under U.S or applicable copyright law Foundation Mathematics for Biosciences At Pearson, we have