when i hear the word quantitation , what comes to mind is the contrast between two famous 19th - century german scientists , alexander von humboldt and carl gauss , as described in measuring the world : a novel ( kehlmann , 2009 ) .
von humboldt was a great prussian explorer who made many discoveries , including electric eels , in his expeditions to south and central america , which included the casiquiare canal , which connects the orinoco and amazon rivers .
a man of extraordinary energy , he was the consummate quantitater  he measured the heights of mountains and the numbers of lice on the heads of natives .
gauss , as surveyor and astronomer for the kingdom of hannover , was also a quantifier , but with a different style .
charged with devising a practical method to survey on a spherical earth , he laid the foundation for a school of geometry that culminated in the work of riemann and the notion of curved space - time in einstein 's general theory of relativity .
the lesson here is that when you have a theory , measurement becomes very important and intellectually challenging .
have all the relevant parameters been measured ? have all the confounding complications been considered ?
does the theory really fit to the data ? how accurately have the underlying quantities ( parameters ) been measured ?
what is a model ? what is the difference ? in biology , we usually distinguish three types of model : conceptual models , mathematical models , and simulations ( bowne - anderson et  al . , 2013 ) .
they all serve the same general purpose : to explain more complex phenomena in terms of more elementary processes or molecular interactions .
this can be in the form of a cartoon or picture that conveys the author 's idea or explanation of his or her results .
but can it really explain the data ? usually , more work is needed to establish the plausibility of a conceptual model .
a mathematical model is the development of the conceptual model into a system of equations .
the postulates are identified ( often in the form of parameters ) , and the implications or predictions are derived by solving the equations . the equations usually describe the average behavior of a large number of interacting molecules or entities  a so - call mean - field theory ( although variances and other statistical properties can often also be calculated ) .
the solutions are then compared with the experimental measurements  the quantitation of the data . if the equations can be solved analytically , all is well and good because this usually indicates that the solution is simple , or at least familiar . if not , numerical solutions are readily provided by such programs as matlab ( www.mathworks.com ) .
the predictions of the model ( with appropriate parameters ) can then be used to quantitatively account for the experimental data .
this is the process of curve fitting , which can be used to ask whether the theory actually fits the data ( i.e. , establish plausibility ) and to measure the underlying parameters and their uncertainties . a simulation is a computer - based imitation of a system . like a mathematical model , a simulation postulates interactions between underlying molecules or entities . unlike
a mathematical model , however , a simulation models individual behavior ( rather than mean behavior ) and often needs to be repeated a large number of times to obtain an accurate measure of the average behavior .
simulations are especially valuable when modeling small - number systems , which can display highly stochastic behavior .
simulations can be used for establishing the plausibility of a conceptual model , and by changing the parameters , the simulator can experiment with the model to gain insight into how it works .
the first is that they often contain many details such as the precise initial conditions or the order of the interactions . in this sense
they are often like experimental results , which also depend on many details ( e.g. , the composition of the buffers ) . in light of well - documented cases in which the results of simulations depended on these details , it has been argued that a simulation should be treated like an experimental result in the sense that it should be confirmed in different laboratories before being accepted ( mitchell , 2009 ) .
a second potential limitation of simulation is that although it can establish plausibility of a mechanism , it might provide only limited insight .
this is especially the case if the system behavior is unexpected and the elementary interactions give rise to so - called emergent or collective behavior . in this case
, the behavior of the simulation may be as mysterious as the biological phenomenon being modeled , even when the simulator can probe the model by varying the model parameters ( analogous to performing wet experiments ) .
an alternative approach is to use so - called toy models ( or idea models ; mitchell , 2009 ) , which , while oversimplifying the system under study , are nevertheless simple enough to solve using a mathematical model , thereby providing insight into mechanism .
an example of a toy model is turing 's reaction - diffusion mechanism of pattern formation ( turing , 1952 ; gierer et  al . , 1972 ) .
i do not want the reader to think that simulations are always inferior to mathematical models . indeed , there are now very good algorithms that have solved many of the technical problems associated with simulations ( e.g. , the gillespie algorithm ; doob , 1945 ; gillespie , 2007 ) .
the physicist freeman dyson recounts attending a conference at which enrico fermi criticized the complexity of dyson 's model by quoting the mathematician john von neumann ( dyson , 2004 ) :  with four parameters i can fit an elephant , and with five i can make him wiggle his trunk . 
actually , it is not so easy to model an elephant with four parameters , although we recently achieved it , even making the trunk wiggle with the fifth parameter ( mayer et  al . , 2010a ) !
if a model has a small number of premises or parameters that give rise to broad and/or unexpected implications , then we might call it a theory .
evolution  the modern evolutionary synthesis  is a theory because it makes a relatively small number of postulates yet predicts a very broad range of observations , including the laws of inheritance and models of population genetics .
it could be argued that other models , especially those underlying spatial oscillations ( e.g. , turing patterns ) and temporal oscillations ( e.g. , the cell cycle ; see later discussion ) , rise to the level of theories .
theories and models can be used to establish plausibility , sufficiency , or impossibility . they can also be used to measure .
the importance of establishing plausibility is illustrated by a recent example in which a textbook conceptual model was shown not to explain the observations .
myosin cortex was driven by a gradient of tension ( bray and white , 1988 ) . however , using experiments ( laser cutting ) and theory , it was found that cortical flow in the one - cell caenorhabditis elegans embryo is driven not by a gradient in tension but by a gradient of contractility ( i.e. , motor activity ) : in the direction of flow , there is no tension gradient , but , counterintuitively , in the transverse direction , there is a tension gradient but no flow ( mayer et  al . , 2010b ) !
this example shows the importance of developing a mathematical model ( and testing it ) to establish the plausibility of a conceptual model .
two examples of the use of theory to establish sufficiency are the action potential and the cell cycle . in a remarkable feat that serves as a foundation for neurobiology , hodgkin and huxley ( 1952 ) formulated and solved equations showing that the action potential in nerve cells can be accounted for by the voltage - dependent interactions between sodium and potassium conductances ( channels in the modern parlance ) .
the second example concerns the cell division cycle . following seminal discoveries showing the necessity of phosphorylation for controlling the transitions between phases of the cell cycle , tyson and novak ( 2011 ) showed the sufficiency of phosphorylation networks to drive the transitions . in other words
, they answered the question ,  what do you need to build a molecular switch 
( answer : feedback and cooperativity ) , and , by doing so , they made the problem of the cell cycle  finite .  in the process , they made key predictions that were confirmed experimentally and they explained many phenotypes .
an excellent example of the use of theory to show impossibility is the prescient work of ken machin ( machin , 1958 ) .
he proved that the periodic bending motion of cilia and flagella ( such as sperm tails ) can not be driven by motor activity localized to the base of the cilium only : if the beating were driven by a whip - like process at the base , the amplitude would die out with a characteristic shape due to the damping from the fluid .
this shape is not consistent with the observed beating patterns , whose amplitudes are maintained along the length of the flagellum .
instead , he proposed motors all along the length , with a traveling wave of activity propagating from base to tip .
not bad , considering that the motor dynein had not been discovered ( gibbons and rowe , 1965 ) , nor was it known whether the filaments inside the flagellum ( the microtubules ) slid or contracted ( they slide ; satir , 1965 ) . of interest
showing impossibility is the most powerful use of mathematical models : it allows hypotheses to be falsified , which is difficult to achieve using simulations . in my work , i have used theory as a tool to measure molecular properties and interactions . for example , we used precise mechanical measurements on hair cells  the sensory receptors underlying the sensation of sound and acceleration  to estimate the force needed to open a single channel , the displacement associated with the opening of the channel 's gate , and the number of channels ( howard et  al . , 1988 ) .
in other work , we used theory to infer the bending stiffness of actin filaments and microtubules from an analysis of their shape fluctuations ( gittes et  al . , 1993 ) .
more recently , we showed how molecular interactions between kinesin motor proteins and microtubules can give rise to length - dependent microtubule depolymerization , which may play a key role in determining the size of organelles such as the mitotic spindle ( varga et  al . , 2006 , 2009 ;
these include the following : 
 morphology : what determines the size , number , and shape of cells and organelles?signaling : how do molecules compute , and what is the role of spatial localization and compartmentalization?scaling : how do we bridge from one scale to another ? are there universal laws ?
morphology : what determines the size , number , and shape of cells and organelles ?
signaling : how do molecules compute , and what is the role of spatial localization and compartmentalization ? scaling : how do we bridge from one scale to another ? are there universal laws ?
especially important is to understand how variability is minimized within a cell to produce reproducible structures and signals , and amplified among cells to enlarge the number of stimuli to which a population of cells can respond .
there has been rapid development of single - molecule techniques , making it possible to manipulate individual molecules using optical and magnetic techniques and to visualize their activity inside cells using fluorescence microscopy . in this way
it is even possible to sequence single molecules of dna ( www.pacificbiosciences.com ) . at the whole - cell level
the developments are as impressive : image processing techniques can be used to segment single cells in tissues to measure shape and activity , their total protein assayed by mass spectrometry , and their genomes sequenced .
these studies have revealed remarkable cell - to - cell variability of  identical  cells .
hand in hand with these developments , we must have theory and models to make sense of the experimental results , as exemplified by gauss , rather than the collection of data for their own sake .
i predict that in the future , theory will drive biology as it does physics and engineering .
 quantitative biology is a hot area , as evidenced by the recent establishment of institutes , graduate programs , and conferences with that name . 
 but what is quantitative biology ? what should it be ? and how can it contribute to solving the big questions in biology ? 
 the past decade has seen very rapid development of quantitative experimental techniques , especially at the single - molecule and single - cell levels . in this essay 
 , i argue that quantitative biology is much more than just the quantitation of these experimental results . instead , it should be the application of the scientific method by which measurement is directed toward testing theories . 
 in this view , quantitative biology is the recognition that theory and models play critical roles in biology , as they do in physics and engineering . by tying together experiment and theory , 
 quantitative biology promises a deeper understanding of underlying mechanisms , when the theory works , or to new discoveries , when it does not . 