Many organisms use cilia to move inside fluids. Which of the following
statements is true for ciliary motion?
The viscous friction coefficient for the motion of a cilium parallel to its axis is
smaller than the viscous friction coefficient for the motion of a cilium
perpendicular to its axis.
The viscous friction coefficient for the motion of a cilium parallel to its axis is
equal to the viscous friction coefficient for the motion of a cilium perpendicular
to its axis.
The viscous friction coefficient for the motion of a cilium parallel to its axis is
larger than the viscous friction coefficient for the motion of a cilium
perpendicular to its axis.
The viscous friction coefficient for the motion of a cilium parallel to its axis tends
to infinity when compared with the viscous friction coefficient for the motion of a
cilium perpendicular to its axis.
The correct statement is: “The viscous friction coefficient for the motion of a cilium parallel to its axis is smaller than the viscous friction coefficient for the motion of a cilium perpendicular to its axis.” This anisotropic friction is fundamental to how cilia and flagella generate effective propulsion in a viscous fluid.
Introduction: ciliary motion and viscous friction
Cilia beat in a low Reynolds number regime, where viscous forces dominate over inertial forces, so propulsion depends sensitively on how drag (viscous friction) acts on a slender filament such as a cilium. In this regime, a cilium experiences anisotropic friction, meaning the viscous friction coefficient for motion parallel to its long axis differs from that for motion perpendicular to its axis. This anisotropy is what allows the effective and recovery strokes to generate a net fluid flow.
Core concept: anisotropic viscous friction of cilia
Hydrodynamically, a cilium or flagellum can be treated as a slender cylinder moving through a viscous fluid such as water or mucus. Slender‑body and resistive‑force theories show that the local viscous friction (or drag) coefficient perpendicular to the cylinder axis ξ⊥ is larger than the coefficient parallel to the axis ξ∥.
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For a given local velocity, motion perpendicular to the axis produces greater viscous resistance, because more fluid must be displaced sideways around the filament.
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Motion parallel to the axis “slices” through the fluid, so the cross‑sectional area presented to the flow is effectively smaller and the corresponding viscous friction coefficient is lower.
This inequality is usually expressed as:
ξ⊥>ξ∥, where both coefficients are finite and depend on filament length, radius, and fluid viscosity.
Option‑wise analysis
Option 1
Statement:
“The viscous friction coefficient for the motion of a cilium parallel to its axis is smaller than the viscous friction coefficient for the motion of a cilium perpendicular to its axis.”
Evaluation:
This statement is true and is the correct answer. The hydrodynamics of flagella and cilia are routinely modeled using anisotropic friction with ξ⊥>ξ∥. Experimental and theoretical work on ciliary and flagellar beating explicitly uses a larger perpendicular friction constant per unit length compared with the parallel component to explain net propulsion and metachronal coordination.
Reasoning in biological terms:
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During the effective stroke, the cilium sweeps largely perpendicular to its axis relative to the fluid, exploiting the higher perpendicular drag to push more fluid and generate transport.
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During the recovery stroke, the cilium changes its configuration and orientation so that motion is more parallel or closer to the cell surface, facing lower effective drag and thereby minimizing backward flow.
Thus, the lower parallel friction coefficient and higher perpendicular friction coefficient are essential to directional ciliary transport, making Option 1 correct.
Option 2
Statement:
“The viscous friction coefficient for the motion of a cilium parallel to its axis is equal to the viscous friction coefficient for the motion of a cilium perpendicular to its axis.”
Evaluation:
This statement is false. If the viscous friction coefficients parallel and perpendicular to the ciliary axis were equal, the friction would be isotropic. In isotropic drag, any reciprocal back‑and‑forth motion of a slender filament would produce no net displacement, as captured in Purcell’s “scallop theorem” for low Reynolds number locomotion.
Why this cannot be correct for real cilia:
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Real ciliary propulsion depends on non‑reciprocal beating patterns plus anisotropic friction; if the drag were the same in all directions, the time‑reversed effective and recovery strokes would cancel each other hydrodynamically.
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Theoretical descriptions of flagellar swimming always introduce distinct parallel and perpendicular resistance coefficients to reproduce observed swimming speeds and flows.
Therefore, equality of the two friction coefficients contradicts established hydrodynamic theory and experimental observations of ciliary motion.
Option 3
Statement:
“The viscous friction coefficient for the motion of a cilium parallel to its axis is larger than the viscous friction coefficient for the motion of a cilium perpendicular to its axis.”
Evaluation:
This statement is false and simply reverses the actual inequality. As described by slender‑body and resistive‑force theory, the friction coefficient in the perpendicular direction exceeds that in the parallel direction for a slender filament in a viscous fluid.
Physical argument:
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When moving perpendicular to its long axis, the filament must push a larger volume of fluid sideways, which generates stronger viscous resistance.
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When sliding parallel to its long axis, the filament displaces fluid in a narrower “tube” around itself and suffers less viscous drag per unit length.
If the parallel coefficient were larger, the effective stroke would not be the high‑drag stroke, undermining the basic mechanism of net fluid transport by an asymmetric beat cycle.
Option 4
Statement:
“The viscous friction coefficient for the motion of a cilium parallel to its axis tends to infinity when compared with the viscous friction coefficient for the motion of a cilium perpendicular to its axis.”
Evaluation:
This statement is false and physically unreasonable. Both ξ∥ and ξ⊥ are finite, determined by the fluid viscosity and the cilium’s geometry (length and radius). Neither coefficient becomes infinite under normal biological conditions; an infinite friction coefficient would imply that any finite parallel velocity requires an infinite force, which is impossible for real cilia powered by molecular motors.
Hydrodynamic context:
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In all realistic models and measurements, the friction coefficients differ by an order‑one factor (often roughly of order 1–2), not by divergent ratios.
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Cilia exhibit substantial motion both perpendicular and somewhat parallel to their axes over the beat cycle; this would not be possible if the parallel friction tended to infinity.
Therefore, Option 4 is clearly incorrect from both mathematical and biological perspectives.
Summary table: friction coefficients and option truth values
| Aspect | Parallel to axis (ξ∥) | Perpendicular to axis (ξ⊥) | True/false option implication |
|---|---|---|---|
| Magnitude of viscous friction coefficient | Smaller for slender cilia and flagella | Larger due to greater fluid displacement | Supports Option 1 (true) |
| Equality of friction coefficients | Would eliminate anisotropy and hinder propulsion | Same as parallel → isotropic drag | Option 2 (false) |
| Parallel larger than perpendicular | Contradicts slender‑body hydrodynamics | Would be incorrectly assumed smaller | Option 3 (false) |
| Parallel friction tending to infinity | Physically impossible for real cilia | Finite, not vanishing in comparison | Option 4 (false) |
Exam‑oriented takeaway (with keyphrase)
For CSIR‑NET/JRF, GATE‑BT, and similar life science exams, remember that ciliary motion viscous friction coefficient is anisotropic, with ξ⊥>ξ∥. This anisotropy allows cilia to generate net fluid transport using a non‑reciprocal beat pattern, making the statement “parallel friction coefficient is smaller than perpendicular friction coefficient” the only correct option.
