biophysics minimals

kinetic energy

kinetic energy

the work done by an object of mass (m) and velocity (v) as its speed approaches 0

E = 1/2mv^2

electron volt

a unit of energy equivalent to the kinetic energy of a single electron accelerated through a voltage of 1 volt

force

a vector quantity characterized by the ability to cause acceleration

acceleration

the rate of change of velocity with time

a = dv/dt, where a is acceleration, dv is change in velocity, and dt is the change in time

newton’s second law

the acceleration of an object is given by the ratio of the net force acting on the object and its mass

a = Fn/m, where Fn is the net force and m is the mass

centripetal acceleration

the rate of change in the direction of velocity over time

a = v^2/r = w^2 • r, where v is velocity, r is the radius, w is angular velocity

angular velocity

the ratio of the angle (delta phi, usually measured in radians) traversed to the amount of time it takes to traverse the angle (delta t)

w = d phi/dt

momentum

the product of the mass (m) and the velocity (v) of an object

p = mv

moment of inertia

• serves the same purpose in circular and linear motion

• characterized by the resistance of the object against angular acceleration

• that of a point-like object can be calculated by the following equation: I = mr^2

angular momentum

• serves the same purpose in circular and linear motion

• by definition, its the product of an object’s angular velocity and its moment of inertia

potential energy of an object in a homogenous gravitational field

the potential energy of an object with a mass (m) at a height (h) in a homogenous gravitational field characterized by a gravitational acceleration of (g) given by the following equation:

Epot = mgh

potential energy of a charged object in an electrostatic field

the electrostatic potential energy of an object with a charge (Q) at position (A) is given by the equation:

Epot = Q • Ua, where Ua is the potential difference at point A

work

the amount of energy transferred by a force

W = Fs, where W is the work done, F is the net force acting on the object, and s is the displacement in the direction of the force

buoyant force

• the upward force exerted by a fluid/gas on an object immersed in it

• its equal to the weight of the fluid/gas that the body displaces (archimedes principle)

• Fb = pf • V’ • g, where Fb is the buoyant force, pf is the density of the fluid, V’ is the submerged volume, and g is the gravity

general form of the work-energy theorem & special forms for electric and gravitational field

general form: KEb - KEa = 1/2 • m • (vb^2 - va^2) = Wab, where m is the mass, va & vb are the speed of the objects at the respective points, Wab is the work done on the object between points A and B

electric field: KEb - KEa = Q • Uab, where Q is the charge of the object, Uab is the electric potential difference between points A and B

gravitational field: KEb - KEa = mgh(ab), where h(ab) is the difference between points A and B

power

• the rate at which work is done

• P = W/t, where P is power, W is work done, and t is time

• the unit of power is Watts (W = J/s)

voltage

• the voltage between points A and B is the difference between the electric potential between A and B

• the unit is volts (V)

• if the difference between A and B is 1V, then the work required to move a charge of 1 coulomb from point A to B is 1 joule

electric current

• the amount of charge transported across a boundary per unit time

• the unit is ampere (A)

• A = coulomb/seconds

resistance

• according to Ohm’s law, resistance (R) of a piece of conducting material is the ratio of the voltage applied across it (U) and the current running through it

• R = U/I

• unit is Ohm (omega = V/A)

electric dipole

• is the separated pair of positive (+q) and an equal amount of negative (-q) charge

• electric dipole moment is defined by:

• p = q • r, where r is the separation distance between the charges

• it is a vector quantity pointing from the negative dipole towards the positive

electric lorentz force

• Fl = EQ, where Fl is the lorentz force, E is the electric field, Q is the charge of the particle

• for positive particles, the force is parallel

• for negative particles, the force is antiparallel with the direction of the electric field

• the particle is accelerated in a linear fashion along the direction of the force

magnitude of the magnetic lorentz force

• Fl = Q • v • B • sin (theta), where theta is the angle enclosed by the velocity vector and the direction of the magnetic field, Q is the charge, v is the velocity of the particle, B is the magnetic field

• rule applied to obtain the direction of the lorentz force: right-hand rule

• the moving charged particle is deflected perpendicularly to both the velocity vector and magnetic field w/o change to its kinetic energy

energy and momentum of a photon

• the energy of a photon of frequency f is hf

• the momentum is hf|c=h|λ, where h is Planck’s constant, c is the speed of light in a vacuum, and λ is the wavelength of the photon

ascending order of the electromagnetic spectrum according to energy

radiowaves < microwaves < infrared < visible light < ultraviolet < x-ray, gamma

visible light

the range of electromagnetic radiation observable by the human eye (approx. 400-750 nm)

limiting frequency

f(max) = eU/h, where h is Planck’s constant, e is the charge of an electron

3 most important mechanism of the absorption of gamma and x-rays

• photo effect

• compton-effect

• pair production

major difference between photo effect and compton-effect

• in the photo effect, all energy of the x-ray or gamma photon is used to ionize the atom & set an electron in motion

• in the compton-effect, only part of the energy is used & a photon with lower energy is released

minimal energy of a gamma-photon needed for pair production

• energy equivalent to the mass of the electron and positron according to the Einstein mass-energy equivalence equation:

• E = (me + mp) • c², where me and mp are the rest masses of an electron and positron respectively, c is the speed of light in a vacuum, E is the minimal energy for a gamma-photon to induce pair-production

why is a heavy nucleus necessary for pair production?

required by law of conservation of momentum

annihilation

• process in which an electron and positron (particle/antiparticle pair) collide

• the total mass-energy of the particle system is converted to the energy of 2 gamma-photons

interference

the superposition of waves resulting in the generation of a new wave pattern

constructive & destructive interference

• constructive is when the amplitude of the resultant wave is greater than that of the individual waves

• destructive is when the amplitude of the resultant wave is less than that of the individual waves

requirement for maximally constructive/destructive interference if 2 propagating waves w/ identical wavelength interfere?

• when maximally constructive interference occurs, if the path difference (ds) between the waves is an integer multiple of the wavelength (λ):

• ds = l • λ, where l = 0, 1, 2, 3,… (this happens when the crests of both waves superimpose)

• when maximally destructive interference occurs, if (ds) = (l+1/2)λ, the crest of one wave superimposes on the trough of the other.

condition for constructive interference for an EM wave with λ diffracted on a crystal with a grating constant of c (angle of incidence = 90º)

c • cosa = l • λ, where l = 0, 1, 2, 3,…n, a = angle of diffraction

transverse and longitudinal waves

• in transverse waves, the displacement of oscillating particles is perpendicular to the direction of propagation

• in longitudinal waves, the displacement is parallel to the direction of propagation

monochromatic light

light is monochromatic if its spectrum consists of only 1 wavelength

special characteristics of laser light

• monochromatic

• coherence in time & space

• small divergence

• high light density

types of interactions laser light has with tissues

• photo-thermal (laserthermy, coagulation, vaporization, carbonization)

• photochemical reactions, fluorescence

• photodissociation

• multi-photon ionization

when is EM radiation coherent?

if it consists of photons capable of forming observable interference fringes

basic phenomena the generation of laser emission is based on

• population inversion needed for light amplification but is only possible in systems with 3 or more energy levels

• stimulated emission needed to give rise to coherent monochromatic light

approximate coherence length of laser & classical light source

10^10 cm for laser & a couple of cm for classic light source

ascending order of transitions according to the energy difference (vibrational, rotational, electronic)

rotational < vibrational < electronic

lambert beer law

log(J0/J) = εcL = A

or

J = J0 • 10^-εcL

where:

J - light intensity after passing through a material with thickness L

J0 - incident light intensity when entering sample

A - absorbance (optical density/extinction

ε - molar extinction coefficient

c - concentration in mol/liter

L - optical path length

what the molar extinction coefficient ε depends on

the type of absorbing material, wavelength of light, temperature, type of solvent & environment

how much does the intensity of light decrease if the absorbance (optical density, extinction) of a solution is 1?

it decreases 10-fold

molar extinction coefficient

absorbance (optical density) of a solution with a concentration of 1M and an optical path length of 1 cm

which wavelength is the characteristic maxima of proteins & nucleic acids

proteins 280 nm, nucleic acids 260 nm

amino acids with reasonably high absorption

Tyr, Trp, Phe

singlet & triplet state

• singlet: number of unpaired electrons is 0 & the value of resultant spin multiplicity is 1

• triplet: number of unpaired electrons is 2 & value of resultant spin multiplicity is 3

possible ways of relaxation of an excited electron in a molecule

• vibrational relaxation

• internal conversion

• intersystem crossing

• fluorescence

• phosphorescence

• delayed fluorescence

• energy transfer to another molecule

fluorescence lifetime definition

the time during which the number of excited molecules decreases to 1/e-times (37%) of its initial value

what is a. scintillation, b. chemiluminescence, c. photoluminescence?

processes where photon emission is elicited by

a. ionizing radiation

b. chemical reaction

c. excitation by photons

fluorescnece quantum efficiency (yield)

• fraction of excited molecules emitting a fluorescent photon

• number of fluorescence photons divided by the number of absorbed photons

• rate constant of fluorescence divided by the rate constants of all possible de-excitation processes

why is fluorescence quantum yield always smaller than 1?

because relaxation from the excited state can be accomplished not just by fluorescence emission

lifetime range of fluorescence

τ = 10^-9 - 10^-7 s

lifetime range of phosphorescence

τ = 10^-6 - 10 s

why is phosphorescence lifetime longer than fluorescence

because phosphorescence is the result of spin-forbidden transitions

why förster type resonance energy transfer is a sensitive method for distance measurements?

because its probability is proportional to the inverse 6th power of the separation between the donor and the acceptor

what förster type resonance energy transfer be used for in biology

measuring inter- & intra-molecular distances

parameters determined using fluorescent measurements

• DNA, RNA, protein & lipid content of a cell/the quantity of any kind of material tagged with a fluorescence label

• permeability of the cell membrane

• intracellular enzyme activity

• membrane potential

• intracellular Ca2+ level

• intracellular pH

• presence & density of cell surface antigens & receptors

• mitochondria potential & number of mitochondria per cell

index of refraction

gives speed of light in a given material according to the following equation:

c = c0/n, where c is the speed of light in the material, c0 is the speed of light in a vacuum, n is the index of refraction

snell’s law of refraction

• light beam is refracted when it travels from a material with a refractive index of n1 into a material with a refractive index of n2 (n1 ≠ n2)

• refraction is described by the following equation:

• sin(a)/sin(B) = c1/c2 = n2/n1, where a and B are the angles of incidence and refraction respectively

• c1 and c2 are the speeds of light in the 2 materials

shortest resolvable distance in a light microscope

approx. 200 nm

how can resolving power of a microscope increased

• by decreasing the wavelength of light

• by increasing the index of refraction of the material between the objective and the object

• by increasing the half angle of the objective

numerical aperture

the product of the index of refraction of the material between the object and the objective (n), and the sine of the half angle of the objective (sin(a)): n•sin(a)

formula of resolving power (f) of a conventional light microscope

f = 1/d = (2n•sin(a))/λ

where:

n = refractive index of the medium between the coverslip & the objective

a = half angle of the objective

λ = wavelength of light

d = minimum distance between 2 points at which they are resolvable

dichroic mirror function in a fluorescence microscope

reflects the excitation light & is transparent for the emitted photons => it separates the excitation & emission light paths

excitation filter function in a fluorescence microscope

transparent only in the wavelength range in which the fluorescent dye can be excited => allows only those photons to reach the sample & excite the fluorescent molecule

emission filter function in a fluorescence microscope

transparent only in the wavelength range in which the fluorescent dye emits photons => only photons emitted by the fluorescent dye reach the detector

imagine aberrations in optical systems

• chromatic aberration

• spherical aberration

• astigmatism

• coma

• curvature of the field of the image

• barrel-shaped & cushion shapes distortion of the image

equation for the relationship between the image distance (i), object distance (o), and the focal distance (f)

1/i + 1/o = 1/f

SI unit of diopter

• D (diopter) = 1/f

• its the refractive power of the lens

• f is the focal length of a given lens

• SI unit: 1/m

2 discoveries that made construction of an electron microscope possible

• an electron can be regarded as a wave & its wavelength is only a fraction of the wavelength of visible light

• an electron beam can be focused with a magnetic field

signals that can be detected during an electron microscopic measurement

• back-scattered electrons

• secondary electrons

• characteristic x-rays

• auger electrons

• absorbed electrons

• cathode luminescence

• transmitted electrons

2 types of electron microscopes

• transmission electron microscope (TEM)

• scanning electron microscope (SEM)

principle of TEM

• a thin (typically 100nm thick) sample is illuminated with an electron beam

• sample scatters a fraction of the electrons (doesn’t absorb the electrons)

• an image is formed from the scattered electrons using magnetic lenses

• the image is characteristic of the electron scattering properties of the sample

principle of SEM

• sample is scanned by a thin electron beam

• secondary electrons induced by the electron beam are detected on a pixel-by-pixel basis

isotopes

variants of a chemical element with a given atomic number whose mass numbers are different

list the isotopes of hydrogen with their mass number and constituents of their nuclei

• hydrogen - mass no. of 1 - composed of 1 proton

• deuterium - mass no. of 2 - composed of 1 proton and 1 neutron

• tritium - mass no. of 3 - composed of 1 proton and 2 neutrons

mass defect of nuclei

• mass defect = the difference between the mass of a nucleus and the total mass of its constituents

• Z: number of protons, A-Z: number of neutrons, where Z and A are atomic and mass number respectively

• dm = ([Z]m proton + [A-Z] m neutron) - m atom

• where dm is the mass defect, m proton is the mass of a free unbound proton, m neutron is the mass of a free unbound neutron, m atom is the mass of the given atomic nucleus

relationship between the total binding energy (dE) & the mass defect (dm) of a given nucleus

dE = dm • c², where c is the speed of light in a vacuum

according to Einstein’s mass-energy equivalence principle

how the binding energy per nucleon changes as a function of mass number

binding energy per nucleon has a maximum at nuclei with mass numbers 55-60 (i.e. Fe)

properties of nuclear force (range, strength, direction)

• have limited range

• effect is negligible at a distance of more than a single nucleon

• independent of charge

• very powerful attractive forces whose magnitude exceeds that of electrostatic forces

what kind of energy level does a nucleon reside on in a nucleus compared to the energy of a free particle?

a bound nucleon has negative potential energy compared to a free particle

types of radioactive radiation & characterize particles constituting them

• alpha radiation consists of helium nuclei

• negative beta radiation consists of electrons

• positive beta radiation consists of positrons

• gamma radiation is an electromagnetic radiation consisting of high energy photons

direction of charges in the atomic number and mass number of nuclei during alpha decay, both types of beta decay, gamma decay and electron capture

alpha decay: -4 in mass no., -2 in atomic no.

-ve beta decay: 0 in mass no., +1 in atomic no.

+ve beta decay and electron capture: 0 in mass no., -1 in atomic no.

gamma decay: 0 in mass no., 0 in atomic no.

why is the beta decay spectrum continuous?

besides an electron/positron, an antineutrino/neutrino is also emitted & the energy released during the decay is shared randomly between the 2 particles

electron capture

• some nuclei are capable of capturing an electron in the K shell decreasing their atomic number by 1

• the vacancy generated in the K shell is filled by an electron in a higher shell

• the transition generates characteristic x-ray and/or auger electron

equation describing number of undecayed nuclei as a function of time (law of radioactive decay)

N = N0 • e^-λt

N0: number of radioactive nuclei at t=0

N: number of undecayed radioactive nuclei at time of investigation

λ: decay constant

t: time

physical meaning of the radioactive decay constant

radioactive decay constant is equal to the inverse 1st power of the mean lifetime of a radioactive nucleus

relationship between the radioactive decay constant (λ) and the half life (T)

T = ln(2)/λ

ln(2): natural logarithm of 2

biological half life

the time period during which half of the initial quantity of the radioactive isotope leaves the living system undecayed due to metabolism, secretion or excretion

effective half life

• time during which the initial activity of a given type of radioactive nucleus decreases to half its original value either by physical decay or metabolism

relationship between effective (T eff), physical (T phys) and biological (T biol) half lives

1/T eff = 1/T phys + 1/T biol

relationship between physical (λ phys), biological (λ biol), and effective (λ eff) decay constants

λ eff = λ phys + λ biol

formula describing attenuation of gamma/x-ray radiation in an absorbing material

J = J0 • e^mew • x

where J0 is the incident intensity, J is the transmitted intensity after passing through the absorber of thickness x, mew is the absorption/attenuation coefficient

attenuation coefficient for gamma/x-ray + the SI unit

the reciprocal of the distance at which the intensity of radiation decreases to 1/e-times (~37%) of the initial value

[mew] = 1/m, where m is the distance

how the intensity of alpha-radiation change as a function of the distance from the source

it is constant in the beginning then suddenly decreases to 0

why alpha & beta(-) radiations called directly ionizing?

• they’re charged, so ionization is caused by direct electrostatic interaction with atomic electrons

• their energy decreases in a series of ionizations along their path