lookatphysics.com PREVIEW A mathematical way to think about biology (main site | handouts) + ■ ■ ■ ■ ■ - | + ■ ■ ■ ■ ■ -

Physical sciences perspectives provide biological insights

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

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Office of Physical Sciences-Oncology

I. Some biological patterns can be interpreted using relationships that explicitly involve only a handful of variables

Quantitative relationships can be reasoned from propositions

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

Numbers

2 Bose-Einstein condensate
3 Visual representations
4 Infinity is not a number

Quantities can be related

Algebra

5 Variables
6 Functions
7 Composite functions
8 Inverse functions
9 Square-root function and $i$

Quantitative relationships can be derived

Geometry

11 Euclidean
12 Sine and cosine
13 Approximating $\pi$
14 Right triangles and trigonometric identities

Summation

15 Sigma notation
16 Introduction to infinite series

Combinatorics

17 Permutations and factorials
18 Combinations
19 Binomial theorem

Change accrued is related to rates and durations of change

Limits

20 Limit of a function
21 Improper (infinite) limits
22 Limits "at" infinity
23 Infinite limits "at" infinity
24 Limits do not always exist
25 $\epsilon$-$\delta$ proof of a limit of a linear function

Quantities can change at quantifiable rates

Differentiation

26 Derivatives and differentials
27 Power rule
28 Chain rule
29 Products and quotients
30 Sinusoidal functions
31 Partial differentiation

Power series representations

32 Second derivative describes curvature
33 Determining power series terms
34 Power series for sine
35 Decimal approximation for $\pi$

Change can be accrued

Integration

36 Area under a curve
37 Rate of change of accrued area is the height
38 Integral over interval equals function evaluated at bounds
39 Change of variables rule
40 Separation of variables
$$\int_{a}^{b} \frac{\mathrm{d}f}{\mathrm{d}x} \mathrm{d}x = f(b) - f(a)$$
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 I

41 Compound interest
42 e to the zero
43 Exponent multiplication identity
45 Andrew Jackson
46 Natural logarithm
47 Integral of 1/x

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

48 Incommensurate periods
49 Practical unpredictability
50 Fundamental indeterminism
51 Illustrating memory-free (Markov) processes

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

Cartoon model of protein dynamics

52 Translation and degradation occur over time
53 Differential equation and flowchart
54 Qualitative graphical solution to differential equation
55 Analytic solution and rise time

Law of mass action

56 Oversimplified derivation of law of mass action
57 Oversimplified cooperativity and Hill functions
58 Bistability
$$\frac{\mathrm{d}R}{\mathrm{d}t} = k x_1 x_2 \cdots x_N$$
[1] w3 Iwasa et al. Molecular Flipbook open-source biomolecular animation software

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

Evolutionary game theory (EGT) I

59 Population dynamics with interactions
60 Tabular game theory (comparative statics)

EGT II: Relating dynamics and statics

61 Evolution resulting from repeated game play
62 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)

Data can be summarized and compared with theoretical probability distributions

Varied data and variability can be summarized

Statistics

63 Probability distributions and averages
64 Identities involving averages
65 Dispersion and variance
66 Statistical independence
67 Identities following from statistical independence

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

Probability

68 Bernoulli trial
69 Binomial distribution
70 Poisson limit

Central limit theorem

71 Stirling's approximation
72 CLT a: Statement of central limit theorem
73 CLT b: Optional derivation (special case)
74 CLT c: Properties of Gaussian distributions

Universality of normal distributions

75 Physics labs
76 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 (given a model) low enough to "signify" rejection of a hypothesis

Uncertainty propagation

78 Sample estimates
79 Square-root of sample size ($\sqrt{n}$) factor
80 Error bar "overlap" criterion
81 Illusory sample size

Sample variance curve fitting

82 Chi-squared ($\chi^2$)
83 Minimizing $\chi^2$
84 Checklist for undergraduate curve fitting

$$\chi^2 \sim M - N_{PARAM}$$
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

Distributions and trajectories can be calculated for probabilistic dynamic systems

Stochastic dynamics

85 Master equation
86 Stochastic simulation algorithm

Poissonian copy numbers

87 Stochastic synthesis and deterministic degradation

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

Linear dynamics in discrete time steps can be analyzed using eigenvectors

Linear algebra I

90 Vectors
91 Operators
92 Solution of teaser
$$\det(\mathbb{A} - \lambda \mathbb{I}) = 0$$

Biological example

93 Cartoon quasispecies model

Vector rotation

94 Euler's number II: Euler's formula
95 Linear algebra II: Rotation matrix
96 Complex eigenvalues

Linear dynamics in continuous time can be analyzed using eigenvectors

Differential equations (DEs)

97 Rough introduction to direction fields and of numerical integration

mRNA transcription-protein translation model

98 Transcription-translation model
99 Eigenvector-eigenvalue analysis
100 Cribsheet of linear stability analysis

Adaptation and the incoherent feed-forward loop motif

102 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

103 Romeo and Juliet
104 Twisting nullclines
105 Time delays
106 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).

Quantitative reasoning can be applied to oncology

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

Introduction to physical oncology

107 Introduction to the PSOC Network

Dynamic heterogeneity can be used to study therapeutic dose-scheduling

Dynamic heterogeneity and the metronomogram

108 Stochastic biochemistry
109 Phenotypic interconversion
110 Metronomogram
$$f_S(\Delta t) > f_P(\Delta t)$$
[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)

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

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

111 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: 122–140 (2012).

(In progress)

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

Eventually, all accessible microstates are equally accessed

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

112 Accessible states are eventually equally visited
113 Notating configurations of a system
114 Distribution of energy between small system and big bath
115 Calculating average properties of small subsystems

$$Z = \sum_{i}^{}{g(E_i) e^{-\frac{E_i}{\tau}}}$$

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

Partition function and expected elongation of ideal chain

116 Introduction to model
117 Hamiltonian and partition function
118 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 depends on the ratio of kinetically accessible volumes of phase space associated with two macrostates

Fluctuation theorem and effective ratio of phase space volumes

119 Microstates of the universe are explored over time
120 Microscopic reversibility
121 Ratio of phase space volumes
122 Kinetically accessible volumes of phase space

$$\frac{P_{C_2 \rightarrow C_1}}{P_{C_1 \rightarrow C_2}} = \frac{\mathrm{size}(C_1)}{\mathrm{size}(C_2)} = e^{-\frac{\Delta Q}{\tau}}$$