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Use quantitative reasoning to develop biological insights

"Excellent site for both basic and advanced lessons on applying mathematics to biology."

- Tweeted by the National Cancer Institute Office of Physical Sciences-Oncology

Protein dynamics
Mass action
Evolutionary game theory
Central limit theorem
Curve fitting
Stochastic dynamics
Poissonian copy numbers

Linear algebra
Vector rotation
Differential equations
Physical oncology
Statistical mechanics

Ideal chain
Macroscopic irreversibility
Power series
Euler's number

I. Aspects of biology can be quantitatively modeled

Quantitative reasoning provides insights into biological dynamics

Rates of change can be derived from propositions and used to predict accrued change in biology


Some quantities in biological systems are hypothesized to change over time in apparently stochastic ways

Stochasticity | Back to top

1 Incommensurate periods
2 Practical unpredictability
3 Fundamental indeterminism
4 Illustrating memory-free (Markov) processes
Protein dynamics

Some stochastic processes can be investigated using deterministic reaction kinetics models,

Cartoon model of protein dynamics | Back to top

5 Translation and degradation occur over time
6 Differential equation and flowchart
7 Qualitative graphical solution to differential equation
8 Analytic solution and rise time
Mass action

Law of mass action | Back to top

9 Oversimplified derivation of law of mass action
10 Oversimplified cooperativity and Hill functions
11 Bistability
[1] w3 Iwasa et al. Molecular Flipbook open-source biomolecular animation software
Evolutionary game theory

not just at molecular scales, but also at other scales, e.g., at population scales

Evolutionary game theory (EGT) I | Back to top

12 Population dynamics with interactions
13 Tabular game theory (comparative statics)

EGT II: Relating dynamics and statics

14 Evolution resulting from repeated game play
15 Relationship between time and sophisticated computation

EGT III: Training and validating population dynamics equations

[2] OA Liao and Tlsty, Interface Focus 4:20140037 (2014)
[3] OA Liao and Tlsty, Interface Focus 4:20140038 (2014)

Noise can be interpreted using quantitative relationships

Data can be summarized and compared with theoretical probability distributions


Varied data and variability can be summarized

Statistics | Back to top

16 Probability distributions and averages
17 Identities involving averages
18 Dispersion and variance
19 Statistical independence
20 Identities following from statistical independence

When a model is proposed, the probability of drawing a value of a statistic can be derived

Probability | Back to top

21 Bernoulli trial
22 Binomial distribution
23 Poisson limit
Central limit theorem

Central limit theorem | Back to top

24 Stirling's approximation
25 CLT a: Statement of central limit theorem
26 CLT b: Optional derivation (special case)
27 CLT c: Properties of Gaussian distributions

Universality of normal distributions | Back to top

28 Physics labs
29 Log-normal biology

[4] w3 Normal distribution§Occurrence, Wikipedia (accessed 2014 October 2).

Subjective criteria can be used to proclaim a value of a statistic low enough to "signify" rejection of a hypothesis

Uncertainty propagation | Back to top

30 Quadrature
31 Sample estimates
32 Square-root of sample size ($\sqrt{n}$) factor
33 Error bar "overlap" criterion
34 Illusory sample size
Curve fitting

Sample variance curve fitting | Back to top

35 Chi-squared ($\chi^2$)
36 Minimizing $\chi^2$
37 Checklist for undergraduate curve fitting

PDF Sample-variance curve fitting exercise for MatLab
[5] w3 $\chi^2$ table from M. Akritas's course at Penn State

Choosing criteria based on commercial objectives

Even though academic tradition is a major contributor to the adoption of criteria for declaring statistical "significance," for example, the arbitrary p < 0.05 criterion, it is possible to choose criteria in a more rational manner. For example, a pharmaceutical company could compare potential research and legal costs with potential profits to conclude that it can only afford to move on to Phase III trials for experimental therapies that rule out a null hypothesis at a p < 0.01 level during Phase II trials.

Guard against using quantitative relationships to misinterpret noise

Statistical analysis can be misunderstood, leading to misinterpretation of data

Misinterpreted as certifications of reproducibility

95% confidence intervals do not necessarily thread the means of their associated parent distributions 95% of the time. Confidence intervals can be meaningless when derived from faulty assumptions.

Misinterpreted as certifications of hypothesis "truth"

Rejecting models having p < 0.1 does not ensure that 90% of surviving models are valid. Human imagination easily contrives a large number mutually incompatible models, each of which is consistent with a given set of experimental data at the p > 0.1 level. This consistency cannot logically guarantee that 90% of the models are valid since, by construction, most of them are false.

[6] PDF Nuzzo, Nature 506: 150-152 (2014).
[7] PDF Goodman, Epidemiology 12: 295-297 (2001).

Probabilistic dynamics can be predicted from probabilistic hypotheses

Stochastic dynamics

Distributions and trajectories can be calculated for probabilistic dynamic systems

Stochastic dynamics | Back to top

38 Master equation
39 Stochastic simulation algorithm a: Specifying reactions
40 Stochastic simulation algorithm b: Time until next event
41 Stochastic simulation algorithm c: Type of next event
Poissonian copy numbers

Poissonian copy numbers | Back to top

42 Stochastic synthesis and deterministic degradation
43 Stochastic synthesis and degradation

Dynamical systems can be analyzed by plotting vectors in state space

Accrued change in linear/ish systems can be studied by finding eigenvectors

Linear algebra

Linear dynamics in discrete time steps can be analyzed using eigenvectors

Linear algebra I | Back to top

44 Teaser: Business projection
45 Vectors
46 Operators
47 Intuitive guess for solution of teaser
48 Solution based on eigenvalue-eigenvector analysis

Biological example | Back to top

49 Intro quasispecies a: Population dynamics
50 Intro quasispecies b: Eigenvalue-eigenvector analysis
Vector rotation

Vector rotation | Back to top

51 Euler's formula
52 Linear algebra II: Rotation matrix
53 Linear algebra II: Complex eigenvalues
Differential equations

Linear dynamics in continuous time can be analyzed using eigenvectors

Differential equations (DEs) | Back to top

54 Direction fields and numerical integration

mRNA transcription-protein translation model | Back to top

55 Transcription-translation model
56 Nullclines and critical points
57 Eigenvalue-eigenvector analysis
58 Cribsheet of linear stability analysis

Adaptation and the incoherent feed-forward loop motif | Back to top

59 Incoherent feed-forward loop
60 Adaptation
61 Eigenvalue-eigenvector analysis
62 Cribsheet of almost linear stability analysis

Eigenvector analysis is just one example of a use of quiver fields


Quiver fields can be used to describe oscillations in terms of DEs, time delays, and excitability

Oscillations | Back to top

63 Romeo and Juliet
64 Twisting nullclines
65 Time delays
66 Stochastic excitation

The ability of a system of DEs to produce solutions that resemble data does not rule out alternative descriptions of dynamics.

[8] PDF Ferrell, Jr., Tsai, and Yang, Cell, 144: 874-885 (2011).

II. Quantitative biology provides insights for human society

Quantitative biology provides insights into health, disease, and treatment

Quantitative reasoning can be applied to oncology

Physical oncology

Physical and biological scientists are working together to develop insights into cancer biology and treatment

Introduction to physical oncology | Back to top

67 Introduction to the PSOC Network

Dynamic heterogeneity can be used to study therapeutic dose-scheduling

Dynamic heterogeneity and the metronomogram | Back to top

68 Stochastic biochemistry
69 Phenotypic interconversion
70 Metronomogram
[9] OA Liao, Estévez-Salmerón, and Tlsty Phys. Biol. 9:065005 (2012)
[10] OA Liao, Estévez-Salmerón, and Tlsty Phys. Biol. 9:065006 (2012)

Statistics of rare events can provide insights into the establishment of metastases

[11] OA Cisneros and Newman Phys. Biol. 11:046003 (2014)

Quantitative biology provides insights into renewable energy

(In progress)

III. The structure of biological systems and the behaviors of systems and their parts influence each other

Organization of parts can influence properties and behaviors that they manifest

System behaviors depend on topological organization of component interactions


Local interactions can generate population dynamics qualitatively distinct from those produced by well-mixed models

Modeling spatially-resolved systems | Back to top

71 Deterministic cellular automata

[12] w3 A. Aktipis's agent-based modeling resources
[13] w3 A. Kaznatcheev's introduction to the Ohtsuki-Nowak transform

The success of a non-spatial model in mimicking aggregate population dynamics does not rule out a spatially-resolved model.

Understanding consequences of organizational structure provides insight into biological systems

Spatial organization of biological heterogeneity affects dynamics

[14] PDF Kerr et al. Nature 418:171--174 (2002)
[15] PMC Macklin et al. J. Theor. Biol. 301: 122140 (2012).

Systems can be organized to decrease sensitivity to some factors at the cost of increased sensitivity to other factors

Properties of parts can influence structures into which they are organized

(In progress)

IV. Microstates of a system can be explored when microstates of the universe are explored

Statistically describe how microstates of a universe are accessed

Eventually, all accessible microstates are equally accessed

Statistical mechanics

Statistics can be calculated for small subparts of much larger universes at thermal equilibrium

If closed universes (eventually) explore all accessible states equally, then small subsystems can be studied using partition functions

Equilibrium statistical mechanics | Back to top

72 Accessible states are eventually equally visited
73 Notating configurations of a system
74 Distribution of energy between small system and big bath
75 Calculating average properties of small subsystems

Ideal chain

At finite temperature, even an ideal polymer chain exhibits "entropic elasticity"

Partition function and expected elongation of ideal chain | Back to top

76 Introduction to model
77 Hamiltonian and partition function
78 Expectation of energy and elongation

Statistical mechanics is used to describe the conformations of DNA

[16] PDF Bustamante et al. Curr. Opin. Struct. Biol. 10(3):279--285 (2000)

Microscopic reversibility implies macroscopic irreversibility

Macroscopic irreversibility

Macroscopic irreversibility depends on the ratio of kinetically accessible volumes of phase space associated with two macrostates

DISCLAIMER: The following video chapter is a simplified introduction that can be used in algebra-based high school physics classes. This is not a replacement for the references below.

Fluctuation theorem and effective ratio of phase space volumes | Back to top

79 Microstates of the universe are explored over time
80 Microscopic reversibility
81 Ratio of phase space volumes
82 Kinetically accessible volumes of phase space

[17] PDF Crooks Excursions in Statistical Physics. PhD thesis, University of California, Berkeley (1999)
[18] OA England J. Chem. Phys. 139:121923 (2013)
[19] w3 England Perimeter Institute Recorded Seminar Archive 139:14090073 (2014)

V. Appendix

Quantities can be used to describe changes with time

Quantitative relationships can be reasoned from propositions


A quantity represents a specified outcome from an ordered sequence of possibilities

Numbers | Back to top

83 Manipulatives and addresses
84 Bose-Einstein condensate
85 Visual representations
86 Infinity is not a number

Quantities can be related

Algebra | Back to top

87 Variables
88 Functions
89 Composite functions
90 Inverse functions
91 Square-root function and $i$
92 Quadratic formula

Quantitative relationships can be derived

Geometry | Back to top

93 Euclidean
94 Sine and cosine
95 Approximating $\pi$
96 Right triangles and trigonometric identities

Summation | Back to top

97 Sigma notation
98 Introduction to infinite series

Combinatorics | Back to top

99 Permutations and factorials
100 Combinations
101 Binomial theorem

Change accrued is related to rates and durations of change


Limits | Back to top

102 Limit of a function
103 Improper (infinite) limits
104 Limits "at" infinity
105 Infinite limits "at" infinity
106 Limits do not always exist
107 $\epsilon$-$\delta$ proof of a limit of a linear function

Quantities can change at quantifiable rates

Differentiation | Back to top

108 Derivatives and differentials
109 Power rule
110 Chain rule
111 Products and quotients
112 Sinusoidal functions
113 Partial differentiation
Power series

Power series representations | Back to top

114 Second derivative describes curvature
115 Determining power series terms
116 Power series for sine
117 Decimal approximation for $\pi$

Change can be accrued

Integration | Back to top

118 Area under a curve
119 Rate of change of accrued area is the height
120 Integral over interval equals function evaluated at bounds
121 Change of variables rule
122 Separation of variables

Calligraphy manifests the fundamental theorem of calculus:
  1. Thicker strokes cover paper at faster rates.
  2. Painted area depends both on length through which and on thickness(es) at which brush was drawn.
Euler's number

Euler's number I | Back to top

123 Compound interest
124 e to the zero
125 Exponent multiplication identity
126 Exponent addition identity
127 Andrew Jackson
128 Natural logarithm
129 Integral of 1/x