Engineering matlab/physics

Monitoring chemical reactions using Magnetic Levitation

Measuring density with a ruler...

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𝑚𝑚�⃗�𝑎 = �⃗�𝐹𝑚𝑚𝑚𝑚𝑔𝑔 + �⃗�𝐹𝑔𝑔 + �⃗�𝐹𝑑𝑑 Object is stationary, if �⃗�𝐹𝑚𝑚𝑚𝑚𝑔𝑔 + �⃗�𝐹𝑔𝑔 = 0

�⃗�𝐹𝑚𝑚𝑚𝑚𝑔𝑔 + �⃗�𝐹𝑔𝑔 = 𝜌𝜌𝑠𝑠 − 𝜌𝜌𝑚𝑚 𝑉𝑉�⃗�𝑔 + 𝜒𝜒𝑠𝑠 − 𝜒𝜒𝑚𝑚 𝜇𝜇0

𝑉𝑉 𝐵𝐵 � 𝛻𝛻 𝐵𝐵 = 0

𝐵𝐵𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 =

0 0

− 2𝐵𝐵0 𝑑𝑑

𝑧𝑧 + 𝐵𝐵0

𝜒𝜒𝑠𝑠 − 𝜒𝜒𝑚𝑚 𝜇𝜇0

4𝐵𝐵0 2

𝑑𝑑2 𝑧𝑧 −

𝑑𝑑 2

= 𝜌𝜌𝑠𝑠 − 𝜌𝜌𝑚𝑚 𝑔𝑔

𝑧𝑧 = 𝜌𝜌𝑠𝑠 − 𝜌𝜌𝑚𝑚 𝑔𝑔𝜇𝜇0𝑑𝑑2

𝜒𝜒𝑠𝑠 − 𝜒𝜒𝑚𝑚 4𝐵𝐵0 2 +

𝑑𝑑 2

Levitation height, as equilibrium:

Measuring density with a ruler...

𝜌𝜌𝑠𝑠 = 4 𝜒𝜒𝑠𝑠 − 𝜒𝜒𝑚𝑚 𝐵𝐵0

2

𝑔𝑔𝜇𝜇0𝑑𝑑2 ℎ + 𝜌𝜌𝑚𝑚 −

2 𝜒𝜒𝑠𝑠 − 𝜒𝜒𝑚𝑚 𝐵𝐵0 2

𝑔𝑔𝜇𝜇0𝑑𝑑

• Density of sample depends linearly on levitation height at equilibrium (as well as other system parameters)

Monitoring reactions through changes in density:

𝑧𝑧 = 𝑔𝑔𝜇𝜇0𝑑𝑑2

𝜒𝜒𝑠𝑠 − 𝜒𝜒𝑚𝑚 4𝐵𝐵0 2 𝜌𝜌𝑠𝑠 +

𝑑𝑑 2 −

𝜌𝜌𝑚𝑚𝑔𝑔𝜇𝜇0𝑑𝑑2

𝜒𝜒𝑠𝑠 − 𝜒𝜒𝑚𝑚 4𝐵𝐵0 2

Monitoring reactions through changes in density:

• ∼350 pmol of –CHO per bead(!) • 𝑧𝑧0 is measured as the center of a ∼100-bead cloud

• Difference between b and g, and g and j is one(!) fluorine atom

Monitoring ‘kinetics’ of chemical reactions:

• Beads taken out of reaction chamber at various points in time

• (Washed) beads levitated; density measured independently

• Dispersion of beads at intermediate points is likely due to polydispersity of bead sizes limiting availability of amine groups

Quantifying the amount of bound protein:

ℎ1 = 𝑔𝑔𝜇𝜇0𝑑𝑑2

𝜒𝜒𝑠𝑠 − 𝜒𝜒𝑚𝑚 4𝐵𝐵0 2 𝜌𝜌𝑏𝑏𝑐𝑐𝑚𝑚𝑑𝑑 +

𝑑𝑑 2 −

𝜌𝜌𝑚𝑚𝑔𝑔𝜇𝜇0𝑑𝑑2

𝜒𝜒𝑏𝑏𝑐𝑐𝑚𝑚𝑑𝑑 − 𝜒𝜒𝑚𝑚 4𝐵𝐵0 2

ℎ2 = 𝑔𝑔𝜇𝜇0𝑑𝑑2

𝜒𝜒𝑠𝑠 − 𝜒𝜒𝑚𝑚 4𝐵𝐵0 2 𝜌𝜌𝑏𝑏𝑐𝑐𝑚𝑚𝑑𝑑+𝑝𝑝𝑐𝑐𝑝𝑝𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 +

𝑑𝑑 2 −

𝜌𝜌𝑚𝑚𝑔𝑔𝜇𝜇0𝑑𝑑2

𝜒𝜒𝑏𝑏𝑐𝑐𝑚𝑚𝑑𝑑+𝑝𝑝𝑐𝑐𝑝𝑝𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 − 𝜒𝜒𝑚𝑚 4𝐵𝐵0 2

∆ℎ = 𝑔𝑔𝜇𝜇0𝑑𝑑2

𝜒𝜒𝑠𝑠 − 𝜒𝜒𝑚𝑚 4𝐵𝐵0 2 𝜌𝜌𝑏𝑏𝑐𝑐𝑚𝑚𝑑𝑑+𝑝𝑝𝑐𝑐𝑝𝑝𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 − 𝜌𝜌𝑏𝑏𝑐𝑐𝑚𝑚𝑑𝑑𝜒𝜒𝑏𝑏𝑐𝑐𝑚𝑚𝑑𝑑 ≈ 𝜒𝜒𝑏𝑏𝑐𝑐𝑚𝑚𝑑𝑑+𝑝𝑝𝑐𝑐𝑝𝑝𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐

Quantifying the amount of bound protein:

∆ℎ = 𝑔𝑔𝜇𝜇0𝑑𝑑2

𝜒𝜒𝑠𝑠 − 𝜒𝜒𝑚𝑚 4𝐵𝐵0 2 𝜌𝜌𝑏𝑏𝑐𝑐𝑚𝑚𝑑𝑑+𝑝𝑝𝑐𝑐𝑝𝑝𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 − 𝜌𝜌𝑏𝑏𝑐𝑐𝑚𝑚𝑑𝑑

𝜌𝜌𝑏𝑏𝑐𝑐𝑚𝑚𝑑𝑑+𝑝𝑝𝑐𝑐𝑝𝑝𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 − 𝜌𝜌𝑏𝑏𝑐𝑐𝑚𝑚𝑑𝑑 ≈ 𝑀𝑀𝑀𝑀𝑝𝑝𝑐𝑐𝑝𝑝𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 − 𝑉𝑉𝑝𝑝𝑐𝑐𝑝𝑝𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑁𝑁𝐴𝐴𝜌𝜌𝑠𝑠𝑝𝑝𝑐𝑐𝑠𝑠𝑐𝑐𝑐𝑐𝑝𝑝𝑐𝑐 𝑃𝑃

𝑁𝑁𝐴𝐴 = 6.022 × 1023 𝑚𝑚𝑚𝑚𝑚𝑚−1

𝑀𝑀𝑀𝑀𝐵𝐵𝐵𝐵𝐴𝐴 = 29 𝑘𝑘𝑔𝑔 � 𝑚𝑚𝑚𝑚𝑚𝑚−1

𝑉𝑉𝐵𝐵𝐵𝐵𝐴𝐴 = 3.35 × 10−26 (𝑚𝑚3)

∆ℎ = 𝑔𝑔𝜇𝜇0𝑑𝑑2 𝑀𝑀𝑀𝑀𝑝𝑝𝑐𝑐𝑝𝑝𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 − 𝑉𝑉𝑝𝑝𝑐𝑐𝑝𝑝𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑁𝑁𝐴𝐴𝜌𝜌𝑠𝑠𝑝𝑝𝑐𝑐𝑠𝑠𝑐𝑐𝑐𝑐𝑝𝑝𝑐𝑐

𝜒𝜒𝑠𝑠 − 𝜒𝜒𝑚𝑚 4𝐵𝐵0 2 𝑃𝑃

∆ℎ 𝑃𝑃

≈ −5 𝑚𝑚𝑚𝑚 𝑚𝑚𝑀𝑀

Predicted:

∆ℎ 𝑃𝑃

≈ −8.6 𝑚𝑚𝑚𝑚 𝑚𝑚𝑀𝑀

Measured:

Reaction-diffusion process in the bead:

• Inside of the bead, protein, 𝑃𝑃𝑏𝑏𝑐𝑐𝑚𝑚𝑑𝑑, diffuses toward the center of the bead, reacting with the immobilized ligands, ∗𝐿𝐿, to form protein-ligand complexes, ∗𝑃𝑃𝐿𝐿

Shapiro et al., J. Am. Chem. Soc. 2012, 134, 5637−5646

Reaction-diffusion process in the bead: 𝜕𝜕𝐶𝐶 𝜕𝜕𝑡𝑡

= − 𝜕𝜕 𝑢𝑢𝑐𝑐𝐶𝐶 𝜕𝜕𝑥𝑥𝑐𝑐

+ 𝐷𝐷 𝜕𝜕2𝐶𝐶 𝜕𝜕𝑥𝑥𝑐𝑐

2 ± 𝑅𝑅 Reaction

Diffusion

• 𝑟𝑟, radial coordinate (𝑟𝑟 = 0 in the center of the bead)

• 𝑡𝑡, time (𝑡𝑡 = 0 when beads placed in solution)

• 𝐷𝐷𝑏𝑏𝑐𝑐𝑚𝑚𝑑𝑑, diffusion coefficient for protein in the bead

• 𝐴𝐴, bead surface area

• 𝑉𝑉𝑠𝑠𝑝𝑝𝑐𝑐, volume of solution

• 𝑁𝑁𝐵𝐵, number of beads

@𝑡𝑡 = 0: ∗𝐿𝐿 = ∗𝐿𝐿 0

; 𝑃𝑃 𝑠𝑠𝑝𝑝𝑐𝑐 = 𝑃𝑃 0,𝑠𝑠𝑝𝑝𝑐𝑐; 𝑃𝑃 𝑏𝑏𝑐𝑐𝑚𝑚𝑑𝑑 = 0; 𝑃𝑃 𝑏𝑏𝑐𝑐𝑚𝑚𝑑𝑑

𝑐𝑐𝑒𝑒𝑐𝑐 = 𝐾𝐾𝑏𝑏𝑐𝑐𝑚𝑚𝑑𝑑/𝑠𝑠𝑝𝑝𝑐𝑐 𝑃𝑃 0,𝑠𝑠𝑝𝑝𝑐𝑐

Bead volume constant: ∗𝑃𝑃𝐿𝐿 = ∗𝐿𝐿 0 − ∗𝐿𝐿

Penetration of the protein into the bead:

Reaction-diffusion process in the bead: • Have to solve numerically to obtain ∗𝑃𝑃𝐿𝐿 𝑟𝑟, 𝑡𝑡

∆ℎ = 𝑔𝑔𝜇𝜇0𝑑𝑑2

𝜒𝜒𝑠𝑠 − 𝜒𝜒𝑚𝑚 4𝐵𝐵0 2 𝜌𝜌𝑏𝑏𝑐𝑐𝑚𝑚𝑑𝑑+𝑝𝑝𝑐𝑐𝑝𝑝𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 − 𝜌𝜌𝑏𝑏𝑐𝑐𝑚𝑚𝑑𝑑

(total amount of protein- ligand complex in the bead, at any point in time)

Shapiro et al., J. Am. Chem. Soc. 2012, 134, 5637−5646

Protein binding kinetics:

Homework assignment:

• Read and comprehend:

– “Measuring Densities of Solids and Liquids Using Magnetic Levitation: Fundamentals” by Mirica et al., 2009

– “Measuring Binding of Protein to Gel-Bound Ligands Using Magnetic Levitation” by Shapiro et al., 2012

• Review last two lectures – exam is coming!